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1971 Semi-Valuations and Groups of Divisibility. Dolores Richard Spikes Louisiana State University and Agricultural & Mechanical College
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SPIKES, Dolores,Richard, 1936- SEMI-VALUATIONS AND GROUPS OF DIVISIBILITY.
The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1971 Mathematics
University Microfilms, A XEROX Company, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED SEMI-VALUATIONS
and
GROUPS OF DIVISIBILITY
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy
i n
The Department of Mathematics
by
Dolores Richard Spikes B.S., Southern University, 1957 M.S., University of Illinois, 1958 December, 1971 PLEASE NOTE:
Some pages may have
indistinct print.
Filmed as received.
University Microfilms, A Xerox Education Company ACKNOWLEDGMENT
The author Is grateful to Professor Jack E, Ohm for his counsel and encouragement during the preparation of this dissertation, and wishes also to express her deep appreciation to Dr, Robert L. Pendleton for his inspiration and confidence, and for his guidance during the e a rlie r stage of development of this dissertation. The author is also grateful to the Ford Foundation for its financial support from 1968 to 1971.
Finally, the author expresses profound gratitude to her husband,
Hermon Spikes, and to her daughter, Rhonda Kathleen, for their patience, understanding and encouragement during the past three years and especial! during the preparation of this dissertation. TABLE OF CONTENTS
page
ACKNOWLEDGMENT ...... — . ii
ABSTRACT - ...... iv
CHAPTER
I Introduction ------1
II Semi-ValuatIons ------9
III Composite Semi-Valuat i o n s ------22
IV V-homomorphisms and Lexicographically Exact
Sequences — 1+5
V A Class of Groups which are not Groups of
Divisibility of Domains ------53
BIBLIOGRAPHY...... 61
APPENDIX...... — ...... — ...... 63
V IT A ...... 76 ABSTRACT
*v Let R be a commutative ring with identity and denote by R and
T(R) the m uttipiicative monoid of regular elements of R and the total quotient ring of R, respectively. There is associated with R a partially ordered Abelian group G, called the group of divisibility #v of R and a mapping w from T(R) onto G, called a semi-valuation of
T(R). (More generally, G is called a semi-value group of w.) The
n - 1 . +. ring D consisting of 0 and a ll elements x e E w (G ) for each positive integer n, is called the semi-valuation ring of w.
There have been instances where interesting examples of domains were constructed by phrasing ring-theoretic problems in terms of the ordered group G, fir s t solved there, and the solution pulled back to a ring R. The main theorem in the pull-back process has been a theorem of Jaffard's which asserts that any lattice-ordered group is a group of d iv is ib ility . More recently, Ohm has given procedures for constructing a large class of groups of divisibility of domains which are not necessarily lattice,, and also a class of ordered groups which are filte re d and are not groups of d iv is ib ility of domains.
This paper gives procedures for constructing a class of groups of divisibility of rings (not necessarily domains) which properly includes the class constructed by Ohm. Toward that end, we fir s t extend the concept of a semi-valuation of a fie ld to rings which may contain zero-divisors. The notion of a composite of two valuations of fields is then extended to the notion of a composite of two semi valuations of total quotient rings (which may not be fields), and the construction of this composite is then related to an exact sequence of semi-value groups. Necessary and sufficient conditions
for this sequence to be lexicographically exact are given.
Suppose that C is a group of d iv is ib ility . Among the groups of
divisibility resulting from our construction are the following:
(1) Certain groups of the form A (+) Cp (lexicographic sum), where
A is a semi-value group satisfying certain conditions of our con
struction, C is free, and Cp is the group C ordered by a specified
submonoid P of C+ ;
(2) Groups of the form [A^ (+) ( (+) (k, + ) ) ] Cp, where Au is any
semi-value group of the fie ld k, C is to ta lly ordered, P = {c e C :
c = Cj+...+cn for a fixed integer n = I, c. > 0 ; or c = 0}, (k, +)
*4* denotes the additive group of the fie ld k, and card I = card (C — P) .
lfO-*A^*B^Cp-*Q is an exact sequence of ordered groups, v
is a semi-valuation of a ring T(R) with semi-value group B, and
8 : B -* C is a V-homomorph i sm, then /3v is also a semi-valuat ion with
semi-value group C. Under the assumptions that the sequence is
lexicographically exact and (P ~ 0) + C P, we give necessary and
sufficient conditions for j3 : B - C to be a V-homomorphism. Finally, we construct a class of groups which are filte re d , but are not groups of divisibility.
v CHAPTER I
INTRODUCTION
If R is an integral domain, there is associated with R a partially ordered group G, called the group of d iv is ib ility of R, and a map w from the group of units of the quotient fie ld K of R onto G; w is called a semi-valuation of K. When R is a valuation domain, G is the value group and w is a valuation. In this case there is a w ell- developed theory which enables one to relate ideal properties of R to properties of G, There have been instances where a ring-theoretic problem was phrased in terms of the ordered group G, fir s t solved there, and then the solution pulled back to R [12] . In addition, this method has been used to construct interesting examples of domains ( [ 1 3 ] ,[* 0 ).
Krull[7] first proved that any totally ordered group is the group of divisibility of a domain. In the more general case, Jaffard [ 5 ] has proved that any lattice-ordered group is the group of divisibility of a domain, but beyond this it seems that nothing had been done, before Ohm's work in [13], in investigating non-lattice groups of divisibility.
Jaffard provides an example in [5] of a filtered ordered group which is not a group of d iv is ib ility . In [13], Ohm gives procedures for constructing a class of groups, not necessarily lattice, which are groups of d iv is ib ility , and a class of groups which are not groups of d iv i s i b iI i ty .
The primary concern of this paper is that of showing that Ohm's procedure and related theorems can be generalized to yield a larger class of groups which are groups of divisibility and to provide additional examples of filte re d groups which are not groups of
1 divisibility. Our general 1zation is not confined to domains. Thus,
Chapter II is devoted to extending the concept of a semi-valuation to rings with zero-divisors. In the case of a field K, there is a one- to-one correspondence between subrings of K and equivalence classes of semi-valuations of K. We lose this one-to-one correspondence by allowing zero-divisors, but regain a uniqueness condition by the
introduction of the concept of a semi-valuation monoid. When confined to domains, this theory reduces to that found in [13].
In Chapter III, we state and prove our main theorems related to the construction of groups of divisibility. The construction in [13]
involves an extension of the notion of the composite of two valuations to that of the composite of two semi-val uations, and, under certain assumptions, relates the construction of the composite to a le x i cographically exact sequence of ordered groups. We show that this construction can be done in a more general setting and give necessary and sufficient conditions for the relation of the construction to a
lexicographically exact sequence of ordered groups. As an application of our main theorem, we obtain the groups of d iv is ib ility of all sub
rings R of the integral closure of (the ring of integers Z
localized at the prime ideal (p) of Z ), where R Z ^ j , in a quadratic extension of the field of rational numbers.
Let C be an ordered group with its monoid of positive elements
denoted by C+, and let P be a submonoid of C+. Let Cp denote the group C ordered by P. Suppose that 0 -*A^B^Cp~tO is lexico graphical ly exact and that (P — 0) + (C+) S P. In Chapter IV, we
determine necessary and sufficien t conditions for the homomorphism jS:B _ C to be a V-homomorphism. F in ally, in Chapter V, we construct
a class of groups which are not groups of d iv is ib ility of domains or of certain rings with zero-divisors.
Preliminaries
We begin with definitionswhich are of primary concern to the work of Ohm, Consequently, most of the following, or references - especially
to [1] and [5] - can be found in [13].
1, Ordered groups, By an ordered group we mean a commutative group
with a partial ordering. The ordered group G is called f j 1tered if
for any a^ , f G, there exists a e G such that a^, G* denotes
the elements of G greater than or equal to zero. The elements of G
form-a submonoid of G, i.e ., G satisfies; 0 c G and x, y e G implies
x + y e G+. An ordered subgroup B of G is an ordered qroup contained
in G such that B+ = (A+) /"\ B, If D = n D^is a direct product of
ordered groups D^, 0 is called the ordered direct product if
D+ = fd f D : d = 0 for all o>]. The ordered direct sum is defined a------sim ilarly. If aQ, a^ , an are elements of an ordered group G, we
def ine the expression aQ = inf^ [a^, . . . ,an) by aQ = inf^ [a^ ....a^}
if and only if a = a for all a e G such that a = a.,.,.,a . o i n
Exact sequences. A homomorphism a of an ordered group A into an
ordered group B i s ca 11 ed an order homomorph i sm i f or(A ) ^ B ♦ of is + + an order isomorphism if o is a group isomorphism and or(A ) = B .
A short exact sequence of ordered groups
(i.i) o_ a - b£ c^ o -f- + + + is called semi-order exact if a(A ) *= Q f ( A ) B and ft(B ) £ C . ( I , 1) is col Ied order exact if ev(A+) » o '{A )/^ B + and $(B ) = C .
The exact sequence ( l . l ) is said to be 1exicographicaiIy exact if
B+ = fb e B: b e cr(A+) or /J(b) > O). A lexicographically exact sequence is order exact.
If A and C are ordered groups, we order the direct sum A(+)C by defining (A(+)C)+ = {(a,c): c > 0 or c * 0 and a = 0}. Thus, if i and p are the usual injection and projection maps, respectively, we have a 1 exicographicai1y exact sequence
0 - A -i* A(±) C £ C - 0 . QT Q The semi-order exact sequence 0-»A-*B£*C-*0 spl its if there exists a commutative diagram
0 - a 2 B ------8 C -* 0
■l 12 '3
A0C £ C - 0 where ij and i^ are the identity maps, i ^ ' s an or£ier isomorphism, and i, p are the usual injection and projection maps, respectively.
Except for an example in Chapter M l , where e xp lic it mention of this shall be made again, the symbol A(+)C w ill always be used to denote the lexicographic sum.
3. V -homomorph i sms. If B and C are ordered groups and ft is a homomor phism, then ft is called a V-homomorph jsm if for any bQ, b j , . . . .b^ e
B, bQ = infgfhi , • • • ,bn) imp! les jS(bo)» in fc{^(bf) , . . . ,^{bn)} . A
V-i somorph i sm is an isomorphism such that it and its inverse are
V-homomorphisms. A V-imbeddinq of B in C is a V-homomorphism which is one-to-one. A subgroup B of C is a V-subgroup if the identity map is a V-homomorphism. We mention below a few properties of 5
V-homomorphi sms found in [13].
(a) I f cv: A -» B and flx B -* C are V-homomorphisms, then flat is al so. of 8 (b) If0-*A-*B-*C“*0 is lexicographically exact, then or is a V-homomorphism,
(c) I f B and C are lattice-ordered groups, then the homomorphism flx B -* C is a V-homomorphi sm if and only if #( infgfbj ,.. . ,bn3) = infcf/3(bl ) , . . . ,8(bn)] .
In Chapter 1 I , we define a semi-valuatioh on an arbitrary commutative ring R with identity. However, we show that we might as well have defined the semi-valuation on the total quotient ring of R, i.e., we prove that any semi-valuation of R can be uniquely extended to a semi-valuation of the total quotient ring of R. Thus, we include below a brief discussion of some facts concerning total quotient rings which w ill be pertinent to our work..
An element r of a commutative ring R is called a regular element
if s r ^ 0 for any s / 0 in R. r is called a zero-divisor if r ^ 0 and r is not regular. We observe the usual conventions with respect to subrings of a ring and ring homomorphisms. In p articu lar, we require that any subring R of a ring R1 with identity to have the
identity of R1 as its identity element, whenever R has an identity.
Let R be a subring of a ring R'. R' is called a total quot i ent
rjnq of R if and only if each of the following two conditions holds:
(1)Every regular element of R is a unit in R1.
(2) Every element of R‘ has the form ab ' where a, b e R and b
is a regular element of R. Every commutative ring with identity has a uniquely determined
Lola I quot ient r i ng.
A unitary R-module M is called R-torsion free if whenever rm = 0
r e R, m / 0 e M, then r is a zero-divisor of R. Thus, it follows
from the definition of a total quotient ring, that the total quotient
ring of a ring R is R-torsion free.
For commutative rings R, R1 with identity and a surjective homomorphism h: R R1, a necessary and sufficient condition that R1
be its own total quotient ring follows from a theorem of KrulI.
First we state some preliminary concepts.
Defini t ion 1.2. An element b of a commutative ring R is said to be a
zero-divisor modulo A, where A is an ideal of R, if there exists an element c of R not in A such that be e A.
Def in i t i on 1.3. An ideal P of R is a maximal prime divisor of the
ideal A of R if P 2 A and P is maximal with respect to containment
in the set [ x e R: xy e A for some y t A ]. P is called a prime
divisor of A if there is a mu 11iplicative1y closed subset S of R
which does not meet A such that PR^ is a maximal prime divisor of AR<.
The following theorem is due to Krull [8J.
Theorem I .4 . Let A be an ideal of a ring R.
Then (a) every zero-divisor modulo A is contained in a maximal prime
divisor of A;
(b) every minimal prime divisor of A is contained in at least
one maximal prime divisor of A;
(c) Denote by A(M) the ideal consisting of {x c R: xy E A for
some y t M}. Then A = {~\ A(M) where M runs over all maximal prime divisors of A.
Prqposition 1.5, Let R, R1 be commutative rings with identity, and let h: R - R1 be a surjective ring homomorphism. Then R' is a total quotient ring if and only if every maximal ideal containing ker h is a prime divisor of ker h.
Proof. Suppose that R1 is a total quotient ring. The maximal ideals of R1 are just those ideals which are images under h of maximal ideals of R that contain ker h. Since every non-unit of R1 is a zero-divisor, a maximal ideal M1 of R1 is also maximal with respect to containment
in the set of zero-divisors of R1, Thus h ^(M1) is a maximal prime divisor of ker h.
Conversely, if every maximal ideal M of ker h is a prime divisor of ker h, then h(M) is a maximal ideal of R1 and is a prime divisor of 0 in R‘ . q.e. d.
We conclude this chapter with a brief discussion of the principal of idealization [ l l ] . Let M be an R-module and Rl the direct sum
R©M as modules. With multiplication defined by (r,m )(rl 1ml) =
( r r 1, rm' + r'm), R1 becomes a ring containing R and M, in which M is 2 an ideal and M = 0. We say that R1 is the ring formed by the principle of idealization. If I denotes the identity of R, then (1,0) is the
identity element of R1. The units of R( = {(a,m): a is a unit of R] and (a,m) 1 = (a 1 , -(a !) 2m).
An element b of a ring R is called a zero-divisor with respect
to a given R-module M if there is a non-zero element m of M such that
bm = 0. Thus the set of zero-divisors of R1 contains the set f{b,m):
b is a zero-divisor with respect to M}, It follows, then, that if every non-unit of R is in the set of zero-divisors of R with respect to M, ihiH R' is a total quotient ring. Thus, for example, if R is a domain and {B.}. is the set of proper ideals of R, M =» © (R/B.), then R‘ = R(+)M, the ring formed by theprinciple of idealization, is a total quotient ring. CHAPTER I I
SEMI-VALUATIONS
Throughout this paper, we assume that any ring under discussion is commutative and has an identity element denoted by 1. A domain will always mean a commutative ring with 1 and without zero-divisors.
The following notation will be used consistently. If R is a ring, denote by R*' the m ultiplicative monoid formed by the regular elements of R with respect to the ring multiplication and identity, U(R) and
T(R) will denote, respectively, the multiplicative group of units of R and the total quotient ring of R.
Definition I I . 1. A semi-valuation w of a ring R is a map from R" to an additive p a rtia lly ordered group G such that the following {hereafter referred to as axioms (i), (ii), (iii)) are satisfied. A (i) w(xy) = w(x) + w(y) for all x, y f R";
(ii) w(*j + x2 +,..+xn) = infw(R*) fwtxj) , w( x2) ,.. . , w( xr ) ] for any integer n = 2, whenever Xj , x2,...,x , Xj + x2+ ' " ' +Xn are 'n R ’
(iii) w(-1) = 0 .
Remarks on d efin itio n I 1.1.
1) Axiom (i) merely asserts that w is a monoid homomorphism, and ,i, hence w(R ) is a submonoid of G.
2) Axiom (ii) is equivalent to the following two axioms:
( ii1) w(xty) = ' nf'w(R*j £w(x) , w(y)} whenever x, y, x**y e R";
( i i11) w(x, + . . .+xn) = in fw(R*) rw(xj) ,... ,w(xn) 3 whenever x,,..,x , x. + . . , + x are in R* I n i n
j -1 n . and S x . + S x. ^ R' U fo) for j = I ,,n . i=l 1 i=j+l 1
9 Proof of remark 2): It is clear that axiom (ii) implies each of the
above axioms. Conversely, suppose that (ii) and ( ii") hold. We show
by induction that (ii) then holds. The case n = 2 is merely axiom ( ii1).
Assume inductively that (ii1) and ( i i 11) imply (ii) for a fixed integer n, it Suppose that xlt ...,x j, Xj + . . -+>
“ infw(R*)fw(xl + ---+xj-l + xj + l + --*+xn+l) >
By the inductive hypothesis,
w{x j +. •. 1 ) — * ^^^( R") _ ] 5 i ^j +1 ^ * * * ■ 1
wCxn+1) } . Therefore w(x,+. . -+xn+]) = infw(R*) M x , ) w(xn+1j-j
j-1 n+1 j-1 n If S x. + S x. = 0 for some j, then E x. + E x;=~xn+l c R ’ *li=l I i=j+lt . . * I * i=lI ^ t*il* i=j+i I* * j-1 n . 1 II thus W (((S x. + ^ X;) + xn+] ♦ Xj) e5 infw(R4) . ... fw(- J-1 ^ x. „ + j. LJ+1 y1 x, x<)
w(xn+i) , w(Xj)} , By the inductive hypothesis, w(xj + . . ^
infw(R*)fw(xl) ’ " - 'w(xn+l)} • j-1 n+1 I f E x. + E x.^RU fo) for each j = 1,..., n+1, then (ii) i=l 1 r=j+l 1 merely becomes the statement of axiom ( ii" ) .
3) In the case of a domain, axiom { ii" ) is satisfied vacuously
and hence, as seen in remark 2), axiom ( i i 1) implies axiom (ii).
The following example shows that in rings with zero-divisors,
axioms (i), ( i i 1) and (iii) do not imply axiom ( i i " ) . Example 11.2. Denote by Z ^ the localization of the ring of integers Z al the prime idea) (2) of Z. Let M = Z /2\/B j, where fB.) is I he set i of proper ideals of ^ ( 2) ’ Let R = Z^2j (^ ) ^ ^e the ring formed by the principle of idealization. As noted in the introduction, R is a total quotient ring and U(R) = {{a,m): a e U(Z
Let Zp denote the ordered group of integers with positive elements p = [0, 2, 3 , 4 , . . . ] . For each a e Z ^ , write a = 3 - r , where neither the numerator nor demonimator of r belongs to the idea] ( 3) of Z. * kl Define a map v: R" -» Zp by v(a,m) = k . Let x = (3 • rj,m^) , k« ^l"^2 kl k2 y = (3 • r2*ra2^ ‘ Then = ^ * r 5r2« 3 * r | m| + ^ • r2m2^ = kj + k2 = v{x) + v(y) . Thus, ax i om ( i) is ver i f ied. v(£-1) )= v( 3° * {“1 )J
0, so axiom (iii) is satisfied. If x, y e R , then x + y is a zero- divisor, so that axiom ( i i 1) is satisfied vacuously.
v ((l ,0) + (1,0) + (1 ,0)) = v (3 ,0) * 1
f inf- 0(1,0), v( 1 ,0) , v(l,0)] =0 , P and hence axiom ( ii" ) is not satisfied.
Proposition 11.3. Any semi-valuation of a ring R extends uniquely to a semi-valuation of T(R).
Proof. Let w: R" -► G be a semi-val uat ion of R. Define w1 by w1 (—) = w(x) - w(y) for x, y e R". We note that w' is a well-defined map on T(R) since w is well-defined and T(R) is R-torsion free. Also, the restriction of w1 to R is w, j, Xj x2 If Xj, x 2 , y] , y2 C R", then w' ({“ )(— )) = w(xjX2) - w(y]y2)
Xj X2 = w(xj) - w(yj) + w(x2) - w(y2) = w' (— ) - w' (— ) . Thus axiom ( i) is 12
verified. If x.,..,, x , x,+,..x c T(R)', then there exists s e R I * ’ n ’ I n such that sXj,...,sx , s(x^ + ...+ x n) f R . Then w1 ( s (x j+ .. .+x ))
= w(s(X j+ .. .+xn)) 2 ‘nfw^R*j Cw( sx | ) »• ■ • »w(sxn) } , by axiom (ii).
Therefore, «(s) + wU , + ...+xn) = i nfw(R*) { w( s) + w(x j),..., w( s) + w(xn) }
implies that wl (xj + . . .+xn) « infw(j(R *)) t w‘ j) »• ■ • .w’ { * n) 3 • Hence w' satisfies axiom (ii). w'(-l) = w(-l) = 0, and therefore axiom (iii)
is satisfied. X - ] If v Is any semi-va1uation of T(R) extending w, then v(—) = v(xy ) =
v(x) - v(y) by axiom (i) and the fact that v(y ^) = ~v(y). Since
x, y e R" and v extends w( v(x) - v(y) = w(x) - w(y) = w1(—) . Thus, W
is uniquely determined by w, q.e.d.
It follows from Proposition 11.3 that we do not get a more general
situation by defining a semi-vaIuation for an arbitrary ring R rather
than for its total quotient ring T(R), i.e., we may as well have
assumed that R = T(R) in d efin itio n I 1.1. We w i11 make this assumption
in many of the propositions and discussions to follow, explicitly
stating so each time.
In case R = T (R ), w(R ) is a subgroup of G, called the semi-value
group of w. Since we are interested in the image of w rather than
all of G, we assume that w(R ) generates G. and thus w is surjective
whenever R = T(R),
Two semi-valuations w, w1 of a ring R = T(R) having respective
semi-value groups G, G1 are called equivalent semi-valuat ions if and
only if there exists an order isomorphism $ from G to G1 such that
^w = w1 - 1 + Let w be a semi-valuation of R = T(R). w (G ) has the following propert ies:
— 1 + -V PI: w (G ) is a m ultiplicative submonoid of R , i.e.,
(1) W-I (G+) £ R * ;
(2) if x, y , € w ' (G+), then x y e w ^G *) ;
(3) 1 P2: (S w ’(G+))n R w ’(G+) for each positive integer n; 1 (E w " 1(G+) = {x + . . . + X ;x e w"1(G+)]) . I i n i P3: -I (T w"1(G+) . -V Defini tion I 1.4. Let w : R -* G be a semi-valuation of R = T(R). - ] + Then w (G ) is called the semi-valuation monoid of w. Suppose that w and w' are equivalent semi-valuations of a ring R = T(R). Let M = w V ) , M1 = (w1) V g+) , the respective semi- valuation monoids of w and w '. Let x e M. Then w(x) e G+ implies $ w(x) = w1 (x) e G,+ . Thus, x $ (w ')-1 (G'+) = M' . Sim ilarly, since $ is an order isomorphism, x e M1 implies x e M. Thus M = M1 . The converse is obvious; therefore, two semi-valuations are equivalent if and only if their semi-valuation monoids coincide. Propos i t ion 11.5. Let M be any subset of a ring R = T(R) satisfying PI through P3. Then there exists a semi-valuation w of R having M as its semi-valuation monoid. Proof. Define a pre-order of R* by (R") = M, i.e., M satisfies I e M, and M . M = fab : a,b e M] .G M, Let U denote the m ultiplicative sub- group of R consisting of units of M. Then R /U is an ordered group with M/U as the monoid of positive elements. V ? #v We claim that the natural map w : R ' -* R*'/U is a semi-va luat ion. Since w is a monoid homomorphism, axiom (i) is satisfied. Also, -1 e U S H, and thus axiom (iii) is satisfied. Suppose that #V — | — | X | x , Xj + ...x e R" and that x^d ,. .. ,x d e M for some d € R". - 1 * We must show that (x. + ...+ x )d e M. Since ( £ M) A R it suffices 1 n , to show that x^d + . . .+xnd e R*. But R" is a group and hence X j+ ...+ x n e R*, d e R' imply (x ^ + .. ,+x^) d ' e H" , Thus axiom (ii) is satisfied. Lemma 11.6. Let H be a subset of R = T(R) satisfying PI through P3. n The set D consisting of 0 and elements x € SM for each positive ! integer n forms a subring of R, n m n m Proof. Let x e DM , y e DM , Then x = S X., y = D y r> where x - , y. 1 1 1 1 1 J m n+m € M. Since -1 e M, -y. e M also. Then x + D (~y.) = x -y e D M . ’ 1 1 1 Therefore, x -y e D, xy = x ^ y ^ . +x n Vm i s a sum of elements of M since x.y, e M implies x.y. e M for each i ,j . Thus xy e D. Since 1 c D by PI, D is a subring of R. Proposition 11.7. PI through P3 are necessary and sufficient conditions for a subset M of R = T(R) to be contained in a subring D of R with M = DflR . H = D' if and only if R is D-torsion free. Proof. Suppose that M is a subset of R satisfying PI through P3, and let D be the subring of R definedin the lemma. Clearly, MG D ^ * * since M R . On the other hand, if x e D f[ R , then x € n (E M) f ) R ^ M, by P2, for some n. Hence M = D f \ r \ 1 * Conversely, if D is a subring of R, then M = D A R" satisfies PI .1, J. through P3: since D is a subring of R, 1 and -1 e D'/A R \ M^R", and n * if x e SM for some n, then x e D, Hence, if x also belongs to R , 1 * n j. then x e D A R" = M. Therefore, (S M) A R " £ H, 1 For the second assertion, assume that M = D . If there exists x € D,y ^ 0 e R such that xy = 0, then x i D“ since D" = M & R . Therefore, x is a zero-divisor in D, and hence R is D-torsion free. j. Conversely, if R is D-torsion free, then M = D/"I R* = D". Def i ni t ion 11.8. Let w be a semi-valuation of a ring R = T(R). The n -1 + subring D of R consisting of 0 and all elements x € E w (G } for each 1 positive integer n is called the semi-valuation ring of w. Examp 1 e 11.9. Let R = ^ (2 )(3 ^ t *ie r ' n9 defined in example 11.2. Then ' s a subring of R. R" s6 ^(2)' s 'nce ^ € ^( 2) * ^ut 2 (identified with (2,0)) does not satisfy P] through P3. The map w : R*v -» R*'/U(Z^) *s a semi-valuation, with {1} as the sumbonoid of positive elements. Thus, w ' (fl]) generates Z^j » an^ ^(2) *5 therefore the semi-valuation ring of w. Given a subring D of R such that D" satisfies PI through P3, D need not be a semi-valuation ring, as the next example shows. Such a ring will always contain a semi-valuation ring, however; namely, the ring generated by D . Example 11.10. Let Q, be the fie ld of rational numbers, X an indeterminate over Q, and {B.} the set of proper ideals of the polynomial ring Q[X]. 16 Let M = (+) Q.[X]/B.. Then R => Q.[X] H, the ring formed by the i principle of idealization, is a total quotient ring and R" = U(R) = j, f (a, m) : a e Q'}. Let D be the subring Z ^ j[ X ] M of R, where Z ^ j is the localization of the ring of integers Z at the prime ideal (p) of Z. Then R = T(D), and therefore D“ = j " (?) M satisfies PI through P3. The natural map R R /U(D) is a semi-val uat ion of R. However, ^ — 1 (X,0) e D and does not belong to £ w (D /U (D )) for any n. The semi- 1 valuation ring of w is Z^j^P) M, Propos i t ion 11.11. Let D be a ring. Then D" is the semi-valuation monoid of some semi-valuation w with semi-valuation ring D if and only if every zero-divisor of D is a sum of regular elements of D. .i. Proof. Suppose that D" is the semi-valuation monoid of a semi- valuation w of a ring R = T(R) with semi-valuation ring D. Then n j + N O U ( x : x e 2 w (G ) for each positive integer n]. Since * -1 + 1 D = w (G ) , every element of D is a sum of regular elements of D, and hence every zero-divisor is a sum of regular elements of D. Conversely, if every zero-divisor of D is a sum of regular elements of D, let w be the semi-va!uation of T{D) having semi-valuation monoid D O T(D)*. Since T(D) is D-torsion free, D f\ T(D)" = D" » w"1(G+) , and thus it follows that D" is the semi-valuation monoid of w with semi valuation ring D, Corol1ary II.12. Let R be a quasi-semi-local ring. Let be those maximal ideals of R, if any, such that R/M. Z /( 2 ) , the integers modulo the prime ideal (2) of Z. Then R is a semi-valuation ring with semi-valuation monoid R if and only if for each j = 1 k there r\ * J exists an integer n(j) such that / I (M./\ E R )£fcM. . i*j 1 I J Proof. Suppose that R is a semi-valuation ring with semi-valuation monoid R . Since the number of maximal ideals involved in finite, f \ . Since R is generated by Rf , x e f \ M. implies that i /j J i*SJ ‘ n(j) * ^ n(j) * , x e £ R for some n(j) . Therefore, (\ (M./O ER )£±M. . i i^j 1 1 ‘ -* ^ n (j) , Conversely, suppose that (M./"-) £ R ){ ± M . . Let »■ ■ • »^|< i ^ j ■* 1 T J be the maximal ideals of R which are not isomorphic to Z /(2 ). Let h : R R/Mj X ....X R/Mn he the canonical map defined by h{x) = (hj(x),..,,h (x)), where h. : R -* R/M. is the canonical mapfor each k i = l , . . . fn. We show first that every element of M. is a sum of • l I k regular elements of R. Let y e f \ . Let h.(y) = y. . Since i=l 1 J J R/M. Z/(2) for j > k, there exist non-zero elements s., t. e R/M. J J J J such that yj = sj ~ *j • Hence h(y) = 0.•••> 1,sk+| >•••.sn)" • n (1 l»t...... t ) is a sum of units of II R/M. . There exists i n i therefore, units x, z, e R such that h(x) = (1,...,1,s^+),...,sn), n h(z)=(l ,.. . , 1 ,tk+] ,.. . ,tn) . Thus, h(y)=h(x)-h(z) impl ies y-x+z e /"\ M (= Jacobson radical of R) . y-x+z = m e Jacobson radical of R implies that y = x-z+m is a sum of units of R, since z is a unit implies that k -z+m Is a unit. Thus, any element of r \ m, is a sum of regular i = l 1 elements of R. Let g : R —• R/Mj x...x be the canonical map defined by g(x) = (9] (x),.. . »9j<(x) ) * where g. : R -• R/M. , i = ,k. By as sump t ion, there exists y. e / l M. such that y. t M., and y. is a sum of regular J ' « 1 J \ J ,h elements of R. g(Yj) = (0 ,.,,,0,1,0,...,0) with 1 as the j component and zeros elsewhere. For any x e R, gj(x) is either 0 or 1; therefore g(x) is a fin ite sum of certain of the g(y.), say, g(x) = g (y.)+ .. .+g(yJ J k Then g(x-y.- . . . -y^) = 0 implies that x-y^-...-y^ € f \ M. . Since the i = l 1 k y. are sums of regular elements of R and every element e / I M. is a J i = l 1 sum of regular elements of R, then x is also a sum of regular elements of R. Corol1ary 11.13. Assume that R is a quasi-semi-local ring, and let j(R) denote the Jacobson radital of R. Then R/J(R) is a semi-valuation ring with semi-val uat ion monoid (R/j(R))"f if and only if R/M w Z/(2) for at most one maximal ideal M of R. Moreover, if R/J(R) is a semi-valuation ring with semi-valuation monoid (R/j(R))“ , then R is a semi-valuation ring with semi-valuation J, monoid R", Proof. Let M^ denote the maximal ideals of R. Suppose that R/j(R) is a Semi-valuation ring with semi-valuation monoid (R/J(R)) , and that R/Mj R/Mg » Z / (2). The element (l,0,,..,0) of R/Mj x R/M^ x..,x R/Mn cannot be written as a sum of regular elements, since (r] | rni) + ■ • rik* * * * * rnk) 'mPlies that r , 2+ . . - + r, k=1 and hence the number of summands, k, is odd. On the other hand, r21+ ' ' ,+r2k = ^ implies that the number of summands, k, is even. Thus, R/J(R) is not a semi-valuation ring with semi-valuation monoid (R/J(R)) , contrary to our assumption. 19 Suppose that R/Mj » Z(2), R/M. & Z/(2), i ? 1. Then, for any non zero element X| of R/M^, i ^ I , we can find non-zero elements s., i . , u., v ., w. c R/M. such that x. = s.-t., x. = u.+v.+w. . Thus, if i ’ i ’ i i i i i * i i l l ’ 0 , t 2 t n) if = 0; ( x , xn)= (l,u2,...,un) + (1 ,v2>... ,vn) + ( 1 ,w2>. . . ,wn) if Xj = 1 . If R/J(R) is a semi-valuation ring with semi-valuation monoid (R/j(R)J , then for any element x of R, the image of x in R/J(R) under the canonical map is a sum of units of R /J( R ), since R/J( R) is a total quotient ring for R quasi-semi-local. It follows, then, that x is a sum of units of R, since U(R/J(R)) ^ U(R)/(l+J(R)) . Corol 1 ary I I . l*t. The following are always semi-vaIuation rings R of some semi-va luat ion w of a ring R ' 2 R, with semi-val uat ion monoid R a) Any quasi-local ring . b) The polynomial ring A[X] , Proof, a) follows immediately from Corollary 11.13. Suppose that P(X) is a zero-divisor of A[X]. Let Q,(X) be any regular element of A[X], Then B(X) = P(X) + Q,(X) is a regular element of A[X] . Therefore, P(X) = B(X) - Q.(X) is a sum of regular elements of A[X] . Def in it ion 11.15. Let w be a semi-valuation of a ring R = T(R). w is said to be an additive semi-valuation if J* i) w(x) < w(y) implies w(x+y) = w(x) , whenever x, y, x + y e R"; i i) whenever x, r ., x+y e R*, y = rj + . . , + r^ t R* , w(x) < w(r.) for all i = l,...,k implies w(x + y) =w(x), Proposit ion 11.16. Let w be a semi-valuation of R = T(R), with semi valuation ring D and semi-val uat ion monoid D#f. Then w is additive 20 if and only if the regular non-units of D generate a proper ideal of D. Proof. Suppose that the regular non-units of D generate a proper ideal V* “ I P of D. Suppose x, y, x+y e R and w(x) < w (y). Then w(yx ) > 0. — ] *sV — 1 — ] Thus yx € P. x, x+y e R imply (x+y)x = 1 + yx f R , Then 1 + yx ' is a unit in D; otherwise, 1 + yx ' e P, yx ' f P imply that 1 e P. 1 + yx * is a unit implies that w(l+yx = w((x+y)x ')= w(x+y) - w(x) = 0, Let w{x) < w(r.) where y = r|+,..+r^ is a zero-divisor of R; then r. x ' e P, for each i = l,.,.,k, implies (r^+,..+r^)x * = yx ' e P. Then 1 + yx ' is a zero-divisor or a unit. If 1 + yx ' is a u nit, then w(l+yx *) = w(x+y) - w(x) = 0 . Suppose that w Is an additive semi-valuation. Let x. e R such that w(x.) >0, i = l,...,n. If r e 0, then e i ther r i s a zero- divisor and thus rx is a zero-divisor, or w(rx.) = w(r) + w(x.) > 0. 'V Suppose that x ,+ ...+ x e D'. We will show by induction that I n w(X|+.. *+xn) > 0 . Suppose that w(xj+x2) = 0 < w(xj) . Then, since w(x|) *=> w(-Xj), w(x^+x2- x^) = w(x2) = 0, contrary to the assumption that w(x^) > 0 . Assume inductively that the assertion is true for a fixed integer n > 2. Suppose that w(x^+.. .+*n+|) = 0 < w(x.) , i = 1 ,. . . ,n+l, If (x2+ .. .+xn+^) is regular, then w(x2+ .. -+xn+]) > 0 by the inductive hypothesis. Therefore, w(xj+{x2+ . . •+xn+i )) > 0 by the n = 2 case. If z = x 2+**,+xn+i 's a zero-divisor, then w(X|+.. •+xn+i"^x 2+ ' • *+xn-H^ = w(xj) = w(xj + +xn+]) - 0 is a contradiction to w(xj) > 0 . Therefore, w(xj+.. .+xn+l) > 0 • Since any element of D is a sum of regular elements of D, by proposition 11.10 and the hypothesis, the proposition is proved, q.e.d. We conclude this chapter with a few remarks concerning the semi- value group of a semi-valuation. If G is the semi-value group of a semi-valuation w of a ring R = T(R), then G is filtered if and only if JL. ^ R*' = T (0)" , where D is the semi-val uat ion ring of w. Moreover, G is sV “V filtered if and only if for each y e R , there exists s e D such that s y € D \ Hence, G is filte re d if any only if R" is the group generated by D'\ With R and D as in the previous paragraph, if R = T(D) and G is filtered, G is called a group of divisibility. If D is any ring, then the ordered group T(D) /U(D) is called the group of divisibility of D. Finally, it is easily seen that if v is a semi-valuation of R - T(R) with semi-value group G, and if is a V-homomorphism of G onto an ordered group G1, then #v is also a semi-valuation of R, with semi-value group G1, CHAPTER 11| COMPOSITE SEMI-VALUATIONS In [13], the notion of a composite of two semi-valuations of fields is treated. As we remarked in the Introduction, the construction of this composite along with related theorems gives rise to a large class of groups which are groups of d iv is ib ility . We summarize this construction and the main results pertaining to it in the next paragraph. Let K be a fie ld , w a semi-valuation of K, and assume that the semi-valuation ring R of w is quasi-local, with maximal ideal M. Let h : R -* R/M be the canonical homomorphism. Let u be a semi-va luation of the residue field R/M, with semi-valuation ring D of u. Let v be the semi-valuation of K, determined up to equivalence, having semi valuation ring R' = h *(D). Then v is said to be composite with w and u. Denote by A, B, C, respectively, the semi-value groups of u, v, and w. Ohm proves that B is a lexicographic extension of the semi- ct 3 value groups A and C; and, conversely, if A £ 0, and 0-*A -*B ,t*C -, 0 is lexicographically exact, and if B is the semi-value group of a semi-valuation v of the field K with ring R1 such that )3v is a semi- valuation, then i) the semi-valuation ring R of w is quasi-local with maximal ideal M C R 1^ R, and ii) A is the semi-value group of a semi-valuation u of R/M [13] . In this chapter, we remove the restrictions that K is a field and R is quasi-local to obtain a suitable extension of the concept of a composite semi-valuation. 22 23 Let J(R) denote the Jacobson radical (intersection of a ll maximal ideals of R) of a ring R. Lemma 111.1. Let R be a ring and 1 an ideal of R contained in J(R). Then the mapping e + I -* e (l + I) describes a group isomorphism from U(R/I) onto U (R )/(1 + !). Proof. I U(R). Then h1 is a homomorphism from the m ultiplicative group U(R) to the multiplicative group U(R/I). If h(x) e U(R/l), then there exists h(x1) such that h(x)h(x') = 1 , or xx1 -lei. Hence xx‘ e 1 + I ^ LJ(R), and thus x e U(R) . Thus h1 is surjective. Clearly, 1 + 1 ker h1 , Suppose that h1 (x) = 1 . Then h (x -l) =0 impl tes that x * I ( I , or x f 1 + 1 . Thus ker h1 ^ 1 + I . t + I = ker h1 implies that 1 + 1 is a subgroup of U{R) . Therefore, we have the canonical map U(R) il U(R)/(T+1) given by e -• e . (l+ l). The commutativity of the diagram U(R) —-----* U(R/1) U (R )/(1+ i) gives us the isomorphism g' : e + I e . (l + l) . q.e.d. An ideal I of a ring R is said to be regular if I contains at least one regular element of R. Lemma I I I .2. Let I be an ideal contained in the intersection of all regular maximal ideals of a ring R. In each of the following cases, 2k ft the mapping e + I -* e . ( l + l ) A R describes a group isomorphism from U(R/I) onto U(R)/(1 + I)n R* : i) Every maximal ideal of R is regular, ii) R has only finitely many non-regular maximal ideals (In particular, R is quasi-semi-1ocal.) and I contains a regular element. Proof. i) If every maximal ideal of R is regular, then the inter section of all regular maximal ideals is J(R). Hence i) is a direct consequence of Lemma I II . 1. ii) Let Mj,...,M be the non-regular maximal ideals of R. If n n A | M. + I ^ M , where M is a maximal ideal of R, then f \ M • 1 I » 1 i 1=1 1=1 implies that M ^ ^ M for some j . Since the M. are maximal, this implies M. = M. But since 1 has a regular element, M must be regular, a contradiction to the assumption that the M. are non-regular . Hence n n A M. + I = R. Then the canonical homomorphism h : R — R//"\M. x R/l 1-1 1 i=l ‘ n ( n is surjective, R/( A M.) A 1 «=» R/ A M. x R/l , and i=l 1 i=l 1 n n u (r/( A i o n 0 « U(R/A M.)x U(R/I) . i=l ' I' i = l 1 n Since ( A M.) A ' ^ ( A M.) f ] if ] M ) = J(R) , where the M are 1 = 1 1 1 = 1 1 o r the maximal ideals of R that are regular, the mapping U(R) U(R/( A M.) A I) is surjective by Lemma I I 1.1. Then the i = l 1 n n composite map U(R) u(R/{ A M.) A 0 « U(R/ A M.) x U(R/I) - U(R/I) , 4 I I I • | I (=1 1 1=1 P where p is the projection map, is surjective. (1 + 1) P\ R *£ :U (R ); otherwise, if m f I and (1+m) e R f, then 25 1+m e M for some regular maximal ideal M implies 1 e M, since m lies in every regular maximal ideal, (i+o n r ' is clearly contained in ker(p i 1 h') . If p i' h1(x) = I, then x e l+l . Thus, since x also belongs to U(R) , x e (1 + 1)0 R ■ The commutativity of the diagram U(R) U(R /I) gives us the desired isomorphism g1 . Lemma I 11 .3 . Let 1 be a common ideal of the rings R ^ R1 , and suppose that I ^ the intersection of all regular maximal ideals of R1. Then (U(R) + I) f) (R ')*£ U(R). J- Proof. Let t e U(R) , y e I such that t+y € (R')" . Since t+y e (R1) , t+y c U(R1) because y belongs to every regular maximal ideal of R1 and t € U(R) £ U(R') . | - t(t+y')' € U^R^ + I * since t ( t+y) c U(R') and y e I , anideal of R1 . Thus, e R, since 1 C. R. Hence t+y e U(R) , q.e.d. For the remainder of this chapter, we fix the following notation. Let w be a semi-valuation of the ring R = T(R) with semi-valuation rinq R . Let I be an ideal of R contained in the intersection of all 3 w w regular maximal ideals of R Let h be the canonical homomorphism of R onto R / I . and let h1 be the restriction of h to U(R ) . Assume w w ’ w ------that h1 : U(Rw) -» U(Rw/l) is surjective and Rw/I = T(Rv/l) . Let u be a semi-val uat ion of R.,/1, withsemi-valuation ring R,, w * u and let R = h '(R ). Let v be the semi-valuation of R = T(R) , v u determined up to equivalence, by the ring Rv ( i . e ., v is the semi- valuation of R with semi-valuation monoid Ryfl R 1 the existence of v is given by proposition 11,2.) , Let A , Bv , and Cw denote the semi value groups of u, v, and w, respectively. Let P = w( I f\ R") L/ 0, and let (C )D denote the group C with P as its monoid of positive elements, W r W Lemma 111.4. U(R ) = h '1 (UfR ))A r * . ------v u 1 w Proof. It is clear that U(Rv) ^ h ' (U(Ru)) f \ Rw" . Suppose that x c h ( U(R )) f\ R ' . Since h1 is onto, there exists x' f R " such Ll W W that xx' e (1 + 0/1 Rw* £ U(Ry) . Then h(x') = e U(Ru) . Thus, x1 6 h . Therefore, x c U(Rv) . Theorem 111.5. Let R w be the semi-valuation rlnq of a semi-valuation w of a ring R = T(R). Let I be an ideal contained in the intersection of all regular maximal ideals of R^ and such that Rw/1 = T(Rw/l) . Assume that the canonical map h : R -» R / I maps U(R ) onto U(R / I ) . r w w r w' w Let u be a semi-valuation of R /I with semi-valuation ring R , and let w 3 u' = h '(^u) . Let v be the semi-valuation of R = T(R), determined up to equivalence, by the ring Ry , Denote by A Bv, and C , respectively, the semi-value groups of u, v, and w, and le t P = w( 1 f\ R") \J 0. Then a) there exist homomorph i sms a , {} which complete the commutative diagram D below and which make the bottom row L exact; Diagram D U(R ) ------■------T(R) 0 L: 0 . (i is the inclusion homomorphism. ) b) Let M1 = the semi-valuation monoid of v and M = the semi- vaiuation monoid of u. Then the following equivalent conditions hold: (1) 2 {beBv : be a-(A*) or 8(b) > 0} ; (2) M' 3 [h'^M U 0)] n R" . c) The following are equivalent: (1) The sequence L is lexicographically exact; (2) M* = [h *1(M U 0 )] H R* ; (3) h~1[ ( N /I) f\ Ru] £ z(R) U I , where z(R) denotes the zero-divisors of R, for every regular maximal ideal N of R . w d) Assume that M' = R and M « R . Then, if R is a domain, • V u ’ u 1 L is lexicographically exact; if zfR^) ^ I and L is lexicographically exact, R is a domain. ’ u Proof, a) U(R ) = ker w S ker v = U( R ) since R Q R implies ' w' v v w r U(Rv) £ U(Rw) ; thus /? is defined canonically and is a surjection since w is a surjection. Since h' is assumed to be surjective, the composite uh1 is a surjection. ker{uh’)=h*(U (R ))/"lR , =U(R)by Lemma 111.4, u w V and ker (vi) = U(R^) C\ U(R ) = U(Ry)» since Ry is a subring of Rw. Therefore, a is defined canonically and is an injection. b) We show fir s t that (1) holds. Let b e B such that b e cu(A +) ' ' v u or 8(b) > 0. In the fir s t case, since uh1 is surjective, there exists x e U(R ) such that a uh'{x) = b. Therefore, uh'(x) € A + . Thus w u h'(x) e R , Since h ^(R ) = R , this implies that x € R * Thus, u u v v v(x) = b € By . In the second case, there exists x e T(R)" such that w(x) = 5(b) > 0 . Thus, x e I /I R " c Rv; thus b = v(x) e B* . (1) => (2 ): Let x e [h ' (M U 0) ] f[ R" . h(x) = 0 impl ies x e I f\ R* . Then w(x) = .8(b) > 0 implies b f B ^ y ( | ) , Hence x e M*. If 28 h(x) e M, then h(x) c UtR^/l) since M £ (R /l) * 83 U(Rw/l ) . Since I is contained in every regular maximal ideal and x e R (hence also x e Rw ) , we must have x e U(RW)• Then uh'(x) = 0 implies auh'(x) = 0 . Thus by (1 ), Qfuh'(x) e By+ and hence x e M* . (2) a> (1 ): Let b € Bv such that b e ct(A^) or /3(b) > 0 . Let x e R" = T(R)*‘ such that v(x) = b. If /3(b) = w(x) > 0, then x e I f\ R . Hence h(x) = 0, and therefore x e h ' (0)/") R ^ M1 by (2). Therefore v(x) = b e B^+ . 1f b e <*(A^+) , then cruh1(x) = v(x) = b , h1(x) e M, and thus x € h * (M) fl R*'£s M‘ by (2). Therefore, x e By+ . c ) ( O => f Z) : Suppose that L is lexicographically exact. Let x e M' . By the lexicographic exactness of L, v(x) = b e ff(^u ) or /3v(x) = /3(b) > 0 . In the second case x e I /) R“. Hence h(x) = 0, and -1 Vr x € h (0) n R . In the fir s t case, /3v(x) = 0 , so x e U(RW) . Then afuh'(x) = v(x) = b e ) implies uh'(x) > 0, and thus h'(x) c M . Thus x e h (M) f) R since M’ ^ R' (by definition of a semi-valuation monoid) . Thus, in either case, M1 £ [h '(M 1 /0 ) ] A R'( . Equal ity follows from b ). Thus, (1) => (2), (2) => (1 ): Now assume that [h ' (M U 0) ] f) R" = M1 . Let b c By+ . Choose x € T(R)" such that v(x) = b. Then x € M1 . Hence h(x) = 0 or h(x) c H , If h(x) =0, then x e I f] R' . Therefore, w(x) = j9v(x) = /3(b) > 0 . If h(x) € M, then h(x) e ^{Rw/ ^ > 5 *nce ^ ~ ^ ’ Since x € R (1 R £ R and I is contained in every regular maximal w w ideal of Rw, we must have x e U(Rw)(because h(x) is a unit of Rw/l ) • Thus uh'(x) > 0 implies h'(x) e A^+ . Therefore aruh'tx) = v(x) = b impl ies b e ar(A +) . Hence, B + C f b e B : b € cr(A +) or /3(b) > 0} . r u v — v u Equality now follows from b). Thus, (2) => (1) . 29 (2) => (3): Let x C h"1[(N/l) f\ R j . Thenx e Ry = h"1^ ) . If x e R'\ then x e RVH ■ nnp 1 les x € M1 , and thus by c) (2) , x € h * (M (J 0) . Hence h(x) = 0 or h(x) € M = Ru f\ (Ry/O^ . On the other hand, x € h 1 [(N/l)/^ Ru] Implies h(x) is a zero-divisor of R /I (since R /l ~ T(R /I)) or h(x) = 0 . WWW Thus, x / R" ~ ( I f\ R ') . Since R = T(R), this implies x e z(R) U 1 . Hence, (2) => (3) . (3) => (2 ): Suppose h 1 [(N /l)/3 Ru] £ z(R) U I . le t x e M1 . If h (x) / 0, h(x) t Ru 0 (Rw/ I ) “ , then h(x) e N/l for some maximal ideal N/l of Ry/I- But then x e z(R), a contradiction, since h1 C R* . Thus, h(x) = 0 or h(x) ( Ru/1 (Rv/') > i.e., x ( h"'(Hl/0)/lR" . Hence, (3) => (2) . d) Suppose that M' = R , M = R and that R is a domain. Then rr v ’ u u J. _ 1 Ru“ = Ru ~ 0 , Suppose x c M' = Rv~ . Since Ry = h (Ru) , h(x) = 0 or h(x) e R “ = M . Hence x e [h * (M U 0) ] f) R" since Rv" = M' ^ R*' . Suppose that z(Ry) <£ I and L is lexicographically exact. I f x 1 is a zero-divisor of R , then x 1 e z{Rw/l ) . I f x is any preimage of x 1, then x i M1 = Rv" by the lexicographic exactness [c){2)] . Thus, x e z(Ry) • By hypothesis, z(Ry) ^ 1 . Thus h(x) = x 1 = 0 . Therefore, Ru is a domain. Corol i ary 111.6. Let Cw be the group of d iv is ib ility of a ring RW£*R = T( RW) , and let I be an ideal of Rw such that I £ intersection of all regular maximal ideals of Rw, Rw/1 = T(Rw/l ) , and such that the canonical map h : Rw - R,^/! maps U(Rw) onto U(Rw/ l ) . Let P = w(R'f /l I ) U 0 . Let be the semi-value group of a semi-valuation u of R^/1 with semi-valuat3on ring Ru, such that h ^ffN/1) ft R ] £ z(R) U I for every regular maximal ideal N of Rw . If Cw is free, then A (J) (Cw)p is a group of divisibility. Proof. Let R = h '(R 1 . and let B be the semi-value qroup of a semi- v u v valuation v of R determined, up to equivalence, by the ring Ry. Then, by 30 Theorem 111.5(a), there exist homomorphi sms cr, 0 such that the sequence 0 — A ^ B ^ C -* 0 is exact . u V w By Theorem 111.5(c), h” [ ( N / l) f ) R^] S. z(R) U • for every regular maximal ideal N of Rw implies that the sequence (a,ft) is lexicographically exact. Since Cw is free, the sequence splits. A lexicographically exact sequence splits lexicographically when it splits. Hence B w A ,© (cw) ■ q.e.d, v u p In [10], Marot generalizes the notions of valuation rings, valuations, and unique factorization (factorial) domains to rings with zero-divisors. By ([1 0 ], Theorem 5.1) the rings having groups of d iv is ib ility order isomorphic to the ordered direct sum of copies of Z are exactly the unique factorization rings, and by ( [TO], Proposition 3.6) the rings having group of d iv is ib ility order isomorphic to Z are those rings R having a unique regular maximal ideal M generated by the prime element 1T of R; in the latte r case, every regular ideal is a principal ideal Rffn . As in the case for domains, R is then called a discrete valuation ring. The requirement that the ideal I of Rw in Corollary 111.6 be contained in the intersection of all regular maximal ideals is ju s tifia b le : for example, the ordered direct product of a fin ite number n = 2 of copies of Z can only be the group of divisibility of the intersection of a finite number n 2 of discrete valuation rings. Suppose, for example, that P is a submonoid of (Z x Z)+ such that (Z x Z) is filte re d . Suppose that there exist kj , j. e P such that kj - k p kj - kj < (Z x Z) , and d = in f 2XZ ^ \ *^2^ 'mPl'es kj - d, k2 - d c P , By Corollary IV .9, if (P~0) + (ZxZ)+ <£ P, then the identity homomorphism I : (ZxZ)p-ZxZ isa V-homomorph i sm. For such a P, w '(P~0) is never a proper ideal of R , where R is a ring having group w w of d iv is ib ility order isomorphic to Z x Z and w : T(Rw) -• Z x Z is the 31 associated semi-valuation. It follows, then, from Theorem 111.13, that if ?£ 0, then A^^^(Z x Z) p is never a group of d iv is ib ility . An example to which Corollary 111.6 applies is any discrete valuation ring Rw containing a regular ideal I such that the canonical homomorphism h : R -♦ R /I maps U(R ) onto U(R / I ) . Let M = R *r be the unique regular w w r w ' w ' W -is maximal ideal of Rw. Then w(Rwtt f \ R^ ) = [j e Z : j = n for some fixed integer n = 1}. Let P = w(RwTrn/n ^ ) U 0 . By Corollary 111.6, if Au is the semi-value group of a semi-valuation u of R /R ?rn with ring R such 3 r w w u that h"1 [(M/I )/"l Ru] £ z(Rw) U 1, where I = Rwffn , then Au(+)Zp is a group of d iv i s ib i 1i ty. The requirement that h ' I!90 R ] £ z(R) U I for every regular maximal ideal N of Rw £ R = T(R) is essential, as the following example shows. Example 111.7. Let P { j C Z+ ; j = n for some integer n > l) U 0, a sub monoid of Z+ , Suppose that A^ is a group of d iv is ib ility . The sequence 0 -*■ A^ A u © Zp Zp -» 0, where i and p are the usual injection and projection maps, respectively, is lexicographically exact. By Theorem l» .7 . P : A © Zp Z is a V-homomorphism. Let v be the semi-valuat ion of a ring R = T(R) with semi-value group A^ .. © V Then pv = w is a semi valuation of R with semi-value group Z. The semi-valuation ring Rw of w has a unique regular maximal ideal generated by a prime element IT of R such that w(ir) = 1. Then w ^(P ~ 0) = (R tt0) /"I R . Let h : R -* R /R .V 1 be the canonical homomorphism and assume that h maps U (Rw) onto U(Rw/RwTTn) . By Theorem 111.13, there exists a semi-val uat ion u of- R /R irn with ' ’ w w semi-value group Au such that diagram 0 of Theorem 111.5 is valid. Let R^ be the semi-valuation ring of u. Suppose that A^ is filte re d , and suppose 3 * that there exist x 1 , y' e R^ such that x'y' = 0. Let x, y be preimages of x* , y' , respect ively. Then, by Theorem I I i .5. c) (3) , x, y C z(R) f\ Ry = z(Ry) . On the other hand, we also have x, y e R . Write x = a*rJ , y = bff1 , where a, b t Rw»r. Since x, y e Z(RV) antl * € > we have that a, b e Z(RW) • Since a, b i M, h(a) , h(b) e ll(Rw/RwTrn) . Because h ', the restriction of h to U(R ) , is onto, there exist t , t. e U(R,,) such that h(a) = h (tj, W a D W a h(b) = h(t, ) . or a => t + m. b = t + m', for some m, m* e R w11 . Thus. d a a ’ w x = t ir* + m*r* , y = t.w 1 + rn'ir1 and since R ffn d R , we conclude that a ’ ' b w v * * • * * t aflJ • t b'T' C Rv ' THen’ ) = w('t awJ) ° J implies v f t ^ ) = (d ,j) > 0. But the lexicographic ordering of A^(+)Zp implies that j = n or j = 0 and d = 0. If j = 0, then x = a e R*r, contrary to assumption. Therefore, j = n, i.e ., x c Rtr11 . Hence h(x) - x' = 0 . Thus, R is a domain. A u u filte re d implies that TfR,,)" = (R./R, .w1) , and since T(R,,) is a fie ld , this . u w w u implies that T(Ru) = R y R ^ ” ' ( I f a e T (RU)" , (a+ff)(( 1+RwO f \ Rw") C U(Rw)/ ( l+R^tr11) f \ Rw (R /R w11)'1, by assumption; for n i4 I, (a+w)» ((1+Rwirn) f\ R j') i T(RU) ). Hence, n = 1 , contrary to the in itia l hypothesis that P ^ Z+ , Thus, we conclude that if A is filte re d and h1 : U(R ) -* ’ u w U{Rw/l ) is onto', then A| Zp is not a group of d iv is ib ility for P £ Z+, P=fjeZ+ :j = n > l}U o . in particular, 0 Z = Z P = { j e Z+ : j = n>l}Uo is not the group of d iv is ib ility of any ring Ry such that T(Rv) = T(RW) , where Rw is * n a discrete valuation ring and the canonical map h : R -* R ,/R. ir maps = r w w w U(Rw) onto U(Rw/Rw»rn) . Thus, Zp is not the group of d iv is ib ility of a domain, (in Chapter V, we determine all the submonoids P of Z+ such that Zp is filtered. There we prove a proposition from which it follows that for any such P Z+, Zp is never a group o f ’divisibi lit y of a domain.) 33 The next example provides us with a group of d iv is ib ility of a ring with zero-divisors which is not a group of divisibility of a domain. Example I I I .8 . Let R be the subring of the fie ld of rational numbers consisting of all elements — , a, n e Z. Let R1 = R [i/2 ] = {a+bi/2 : 2n a, b e R, i = A T ) . Let a+bi/2 f R1 , We may assume that a, b € Z. Suppose that b / 0 . Then, if a bi/2 e U(Rl), (a+bi/2) ' = a +2b Thus, if a+bi/2 e U(R '), then a + 2b = 2n, for some n e Z . Write a = 2mr , b = 2-^5, where r and s are odd . Then a2 + 2b2 =» 22mr2 + 22j + ^s2 . Suppose 2m < 2j + 1 (Equality clearly cannot hold). Then a2 + 2b2 = 22m(r 2 + 22j*+1"2m . s2) . But r odd, 2j + 1 * 2m imply that r2 + 22** + ^ 2m . s2 is odd. Hence a2 + 2b2 ^ 2n unless r2 + „2j+l-2m 2 jt . , . , . . 2 > , , 2 J . s = 0, and, since b 0, then s ^ 0; thus s = 1 , and 2 hence r = 0 . We reach a similar conclusion in case 2m > 2j +1 . Thus, if a + bi/2 e U(R '), then a = 0, b = ± 2-* . Conversely, since 2n e U(R) for all n, ± 2ni/2 is invertible in R1 for all n. Thus U(R') is generated by i/2. Let {B.} = the set of proper ideals of R1 , M = C ^R '/B . . Let ' i ' D = R1 M, the ring formed by the principle of idealization. Then U{D) = {(a,m) : a e U(R1)} . The subring = Z[i/2] M = {(a+bi/2, m) : a, b f Z, m c H] of 0 has (±1,m) as its only units, since ± 2n is not invertible in Z for n ^ 0, and from the fact that f(± 2n i/2, m) : n e Z) = U(D) 2 U (Z [i/2 ] ( j ) M) . The isomorphism U(D)/U(Dj) -*■ Z described by ( i/2)n . u(Dj) n i s clearly an order isomorphism onto Z, since the regular elements of D| are the elements ((i/2)n, m), n ^ 0 . Similarly, the subring = Z[2i/2] (+) M of 34 D has (±1, m) as its only units, and the elements ((.i/2)n, m) , n = 0 or n = 2 as regular elements. Thus ( i / 2 ) n -* n describes an order isomorphism from Ll( D) /U (D^) onto Zp, P = {0, 2, 3, 4 , . . . } , i . e . , Zp is, up to order isomorphism, the group of d iv is ib ility of . By Example 111.7, Z is not the group of d iv is ib ility of a domain. We P note that the canonical map h: Dj -* / ( ( i/2 ) (+ )h) does not map U(Dj) _ ^ onto U tD ^ O /a Q M ) );fo r example, h(1 + iv/"2) is a unit, but has no p re image in U(Dj) . Corollary 111.6 is particularly applicable whenever is a quasi-semi-1ocal domain, for in this case, if I is any ideal of R , then the restriction h1 of the canonical map h: R -* R /I to U(R ) is onto r w w w U(Rw/l ) , and if, furthermore, I ^ J ( R w), then R^/1 = T(Rw/l ) . Then, if u is any semi-valuation of f*w/l such that the semi-valuation ring R of u is a domain, and if the group of d iv is ib ility C of R is free, u 7 w w then by Corollary 111.6, A^ (+) (^w)p is a group of divisibility, where is the semi-value group of u and P = w( 1 f \ R^*') U 0. In Corol1ary 111.10 below, we apply Theorem I I I .5 and Corollary 111.6 to obtain the groups of d iv is ib ility of subrings of the integral closure of containing Z^j in a quadratic extension of the field of rational numbers. Unless otherwise stated, the notation A x B wi11 be used to indicate the direct product of groups, and the product ordering w ill not be assumed. A x B is to be considered as an ordered group, however, whenever the ordering is explicitly stated. Corollary 111.10 is a consequence of the following more general Corollary 11 1.9. (References preceded by the le tte r A refer the reader to the Appendix.) 35 Corol1ary 11i.9. Let R be a discrete rank one valuation domain with principal maximal ideal Rp and quotient fie ld K. Let R1 be the integral closure of R in a finite extension L of K, and let w be a semi-valuation of L having ring R1 and semi-value group C. Then there exists a semi-valuation of R;/R'pn with semi-value group and homomorphl sms ^ : An x C -* x C such that {An x C)+ = f(a, c+w(pn_1) ) , c > 0; (a, w(pn"j ')) , a e ker{ B2 . j = 0 ,1 ...... n-2; (0,0)] and A^ x C is the group of d iv is ib ility of the ring R + R'pn . Proof. Let R be a discrete rank one valuation ring with maximal ideal generated by a prime element p of R. Let K be the quotient fie ld of R and let R1 be the integral closure of R in a finite extension L of K. By A.11, the field R/(p) can be identified with a subfield k of R'/R'P by the mapping a + {p) -* a + R'p . Let Uj be a semi-valuation of R'/R'p determined, up to equivalence, by the fie ld k. Let h be the canonical map from R1 onto R'/R'p, and let Rj = h ^{ k) . Denote the semi-value group of u^ by A j , and let Bj be the semi-value group of a semi-valuation v^ of L having ring Rj . Let C be the semi-value group of a semi-valuation w of L having ring R1 . By A .12, C is order isomorphic to the to ta lly ordered group of integers Z or to the ordered direct product of <= 2 copies of Z. Hence C is free. By Corollary I 11.6, Bj = A^ C is a group of divisibility. 4* Since the semi-val uat ion ring k of u I s a fie ld , Aj = 0 . By the lexicographic ordering of A^ (+) C, then, Rj is a local ring with maximal ideal R'p . Moreover, by the commutativity of the diagram 36 R' — -----> R'/R'p (A. 1 I) , R ------> R/(p) R £= h ^(k) . Thus R + R'p ^.R| . On the other hand, (R+R'p)/R'p R/R /I R'p by A. 11, and R /") R1 p = Rp . Hence, R /R /lR 'p = k, and therefore, h ^ ( k) = R + R'p . Thus, Rj = R + R'p . Assume the following inductively; n™ I i) Bn is a semi-value group of a semi-valuation of (R+R'p )/(Rp+R'p ) with semi-valuation ring (R+R1p°)/(Rp+R'pP) ; ii) A^ is a semi-value group of a semi-val uat ton of R'/R'p11 with semi valuation ring (R+R'pn)/R 'pn ; iii) There exist homomorph i sms a , B such that 0 -* B A xC ^ A . r n’ 'n n n n-1 x C -* 0 is lexicographically exact ; (An x C)+ = {(a , c+w(pn~l ) ) , c > 0; (0,0); (a, w(pn J' *)) , a e ker(/32+j. . . .,8n) , j = 0, 1 ,...,n - 2 ) . Let be the semi-value group of a semi-vaiuation of (R+R'pn)/(Rp+Rlpn+') with ring (R+R'p'^J/tRP+R'p0 S . Let h : R + R'pn - (R+R1pn) / (Rp+R1pR+^) be the canonical homomorphism. Then h ' t^+R'pn+1)/(Rp+R'pn+1) ] = R + R'pn+' . Let B 'n+j he the semi-value group of a semi-valuation of L having ring R+R'p . By Theorem 111.5, there exist homomorphisms “ n+j > 8n+| such that the sequence n - u 1 * w .l l 0 - V l ------B«1 — (An * C'v n(R'pn ~ 0) U fO) - 0 15 ,e’ graphically exact . Let An+| be the semi-value group of a semi valuation of R'/R'pn+^ with ring R + R'p11*' , By Theorem 111.5, there 37 n+1 exist homomorphisms a ^+p such that the sequence 0 -* ^R+| ------^n+1 B1 , ------C -* 0 is exact. Since C is free, the sequence spl its. n+1 From the inductive hypothesis, (An x C)+ = ((a , c + w(pn_1)} , c > 0; (0,0) ; (a, w(pn~J“ ')) , a e ker(/S2+j. .. .Bn) , j = 0, l , . . , , n - 2 } Therefore, vn( Rn p) = £(a, c + w(pn) , c > 0; (0, w (a, w(pn j ) ) , a e ker(B2+j. .. .£n) , j = 0, !,...,n-2] Hence, we conclude from the lexicographic ordering of (“ n+j , $n+i) that (An+1 x C)+ = {(a, c x w(pn)), c > 0; (0,0); {a, w(pn"j)), a e ker(B2+j . . . ^ n+1) , j = 0, 1 n -1} . Corol 1 arv 111.10. Let Z ^ denote the localization of the ring of integers Z at the prime ideal (p) of Z, and let R1 denote the integral closure of in a quadratic extension of the field of rational numbers. Then, the following are groups of d iv is ib ility of subrings D of R1 containing Z^j . 1) I f D = Z ^ j + R'pn, and R'p= P, where P is the only maximal ideal of R1 lying over ( p ) , then G = Z/(p+l) Z, (lexicographic sum), p ^ 2, n = 1; G = Z/(pn ') x Z/(p+l) x Z, p "= 3, n = 2, with G+ = {(a, b, c): c = n; (0,0,0); (a, 0, n-j-1), where a € (pJ) / ( p n_I) , j = 0, 1 ,. . . , n-2} ; G = Z/(2 n"2) x Z/(2) x Z / ( 3) x Z, p = 2, n = 2, with G+ = (a, b, c, d):d= n; (0 ,0 ,0 ,0 ); (a, b, 0, n-T); (a, 0, 0, n-j-I), where a e (2J' 1)/(2n 2) , j = 1 ...... n-2} 38 2) If D = + R'p", and R'p = P2 , where P is the only maximal ideal of Rl lying over (p),then G = Z / ( p l O Zs , P = 2, n = 1 , S = fO, 2, 3 , . . . ) c i Z+ ; G = Z /(p n) x Z, p = 3 or p = 2 and P = R'tt where w2 e Z, n = 2, with G+ = {{a, b): b = 2n; (0 ,0 ); (a, 2 ( n - j - l ) ) , where a e (pj + ,) /( p n) , j = 0 ,1 ...... n-2} ; G = Z /(2n ') x Z/(2) x Z, p = 2 and P = R'tt where ft I Z, n = 2, with G+ = f(a, b, c,): c ^ 2n; (0,0,0); (a, 0, 2,(n-j-l)) , where I € (2J)/(2n"1), j = 0,1,....n-2} . 3) If D = Z ^ j + R'pn . and R'p = P ^ P 2, where Pj and are t*16 maximal ideals of R' lying over ( p ) , then G = Z/(p-l) (+) (ZxZ)s, p = 3, n = 1, S = {(a,b)e(ZxZ)+ such that a, b = I or a = b = 0} ; 1 1 G = (ZxZ)s, p = 2, n = 1 , S =. {(a.b): a,b ^ 1 or (0,0)] ; G = Z/(p n ') x Z /(p—1) x (ZxZ), p = 3, n = 2, with G+ = {(a, b, c, d); c, d = n; (0 ,0 ,0 ,0 ); (a,0,n-J-l „ n-j-1), where a e (pJ')/(p n S , j = 0 , 1 , . . . , n-2} ; G = Z/(2n-2) x Z/(2) x (ZxZ), p = 2, n = 2, with G+ = f(a,b,c,d) : c.d^n; (0,0,0,0); (a,b,n-l,n-l,); (a,0,n-j-1, n-j-I), where a e (2J 1) / ( 2 n 2) , j = f ...... n-2} , Proof. By A. 12, there exists a subring of R' of the form Z ^ j + R'pn for each prime p and for each integer n = 1, in each of the cases l ) , 2) and 3) above, and these are all of the subrings of R1 containing 39 Z(p) ‘ Hence it suffices to determine, up to order isomorphism, the groups A of Corollary I I I . 9. Then ker(^0 .B ) is easily determined n 4+ j n for each j = 0, 1 ,,..,n-2. By A.11, (Z(p) + Rlpn)/R 'Pn ^ z(p)/(R,Pnfl Z(pJ) = 2(p)/(pn) « Z/(pn). Thus, by A.7, [ ( z( p) + Rl pn)/R ‘ Pn]" tv ^ ( pn) ^ ** Z/(P-1) x Z/(P° , if P - 3 ; Z /(2n 2) x z/(2) , if p = 2 and n ^ 2 ; {o} , if p = 2 and n = 1 . Ar = (R'/R'pnr/U [(Z ^ + R‘pn)/R1 pn] . Thus, it follows from A .13 and A.7 that A^ sa Z/(p+l) x Z/(pn"') , p ? 3, R'p = P ; 2 /(3 ), P = 2, n = 1, R'p = P ; Z/(2n"2) x 2/(2) x 2/(3), p ^ 2, n ^ 2, R'p = P, Z/(pn) , R'p = P2, p = 3 or p = 2 and 2 P = R'tt with n e z^p^ ; z /(2 n" ]) X Z /{ 2 ) , p = 2, and R'p = P2 and P = R'tt with ir2 4 Z^pj ; Z /(p -l) x z A p " " 1), p ^ 3, R'P = P,P2 ; Z /(2 n“2) x Z /(2 ), p = 2, n ^ 2, R'p = P1P2 ; fo ] , p = 2, n = 1, Rl p = P]P2. By A.12, the group of d iv is ib ility of z ( p) 's ° f index one in the group of d iv is ib i1ity of R1 in case R'p = P, and is of index two in case R‘ p = P2 1 f R'p = PjP,, , then the group of d iv is ib ility of R1 is the ordered direct product of two copies of the to ta lly ordered group of integers and the index of the group of d iv is ib ility of Z^pj in each copy is one. Hence, if w is the semiva1uation of T(R') onto the group of d iv is ib iIity of R1 , we have that w(p) = 1, if R'p= P; w(p) = 2, if R'p= P2; w(p) = (1 ,1 ), if R'p = P]R2 . The result now follows from Corollary I I I .9 . - 40 Propos j t ion 1II. 11. Let K be a field, and let v, w be semi-val uat ions of K with respective semi-value groups Bv and Cw. Let Ry be the ring of v. I f 0 -* A -• B ♦ Cw — 0 is a lexicographically exact sequence of semi-vatue groups, then the sequence sp lits if and only if there exists a set M = fx ] S. K* , c e C . such that w(x ) = c and {x * x ,)/ L cJ ’ w’ c c d xc+d 6 U Corollary liI.12. Let be any semi-value group of a semi-valuation u of a fie ld k. Let C be a to ta lly ordered group, and let P = fc e C : c = Cj+...+cn, for a fixed integer n - 1; c. > 0) U 0 . Then, [Ay (+) ( (+)(k, +) )3 (+) Cp is a group of divisibility, where U ' + cardinality of I = card in ality of (C ~ P) and (k, +) denotes the additive group of the fie ld k. Proof. Let (2^ (C) be the group algebra of C over k with quotient field K. By Krull's Theorem (A.3), there exists a valuation w of K with valuation ring R = k + M, where M is the maximal ideal of R. Then k = (k+Mn)/M n ^ R/Mn . Let u be a semi-valuation of k with ring D. Then, by A,1, [(k+M)/Mn]*AJ(D) (k+M)*/U(D+Mn) « k*/U(D) x (l+M)/{l+Mn) . Let c e w(Mn ~ 0) and choose x e K such that w(x) = c. Then n r \ > x e M implies x = 2 x . p . ,x. , where x . . f M . Then w(x) = i = l ln infQW{xj|...Xjn) )...,w(xr|...xrn)} • Since C is totally ordered, we may assume that w ( x j p . . x jn) = w(xf p . . x (n) , . . . ,w(xr p . . x ^ ) . Then w(x) = w(x^ j . . . Xj ) implies x = t X p . - . x ^ , where t e R. Hence w(x) = w(t Xp) + w(x]2)+...+ w(*ln) . Since w(x.j) > 0 , w(x) e P. Now let c € P ~ 0 . c = Cj + ...+ c where c. > 0 . Let x. e K such that w{x.) = c. . Then c e w(Mn 0) . Hence w(Mn ~ 0) = P ~ 0 . Let Au be the semi-value group of the semi-valuation u. Then A » k''/U(D) . By A.6, (l+M)/(l+Mn) sa © (k,+) , where I = cardinality U + 1 of (C ~ P) . Let u1 be the semi-valuation of R/Mn with semi-vatuation ring D. Then the semi-value group A^, of u1 is order isomorphic to A (k ,+ )), by A. 1, where (k ,+ )+ = f 0J Let h : R R/Mn be U 1 -1 the canonical homomorphism and let Ry = h (D) . Let v be the semi valuation of K determined, up to equivalence, by the ring R and let B be the semi-value group of v. By Theorem 111.5, there exist homomorphisms a, B such that the sequence 0 -* A © ((p ( k,+) )-* B ^ Cp -* 0 is lexicographically exact. Since w(x ) = c for each generator x of c c the algebra^ (C) , the xc trivially satisfy the conditions of Proposition 11.11. Hence, the sequence (a ,$) splits, and therefore splits lexicographically. Example 111.13. Let C be the lexicographic sum of the to ta lly ordered group of integers Z and a totally ordered divisible group G (By [6], Theorem the latte r is isomorphic (group) to copies of the additive group of the rationals.). Let (a,b) c C such that (a,b) > 0, b ( G ~ 0. Then, for each integer n ^ 0, there exists h c G such that nh = b (definition of divisible group). Thus, (a,b) e nC+ = {c e C : c = Cj+. . .+cn, c. > 0} for all n # 0 . For n > 1, (n-1 ,0) e C+, (n-J,0) ^ nC+ . Thus, cardinality C+ (nC+U 0) = n ■ 1 . By Corollary 111.12, if A^ is a semi-value group of a field k, then [Au Ck.+))] © Cp, P = nC+U[0) , is a group of divisibility. In particular, if k is a fin ite fie ld G F(pm) , then by A .10, the only subrings of k are subfields G F(pr) where r divides m. Hence, if q = (pm -l)/(pn -1), then [Z/(q) Q ( (+) Z/(p))] @ Cp is a group 1 of divisibi1ity. 42 More generally, if C is the lexicographic sum of a totally ordered free abelian group Gj and a totally ordered divisible group G 2 , then [\ © ( © (k,+))] © (c, © GjJp is a group of d iv is ib ility , where P = nC+ U Co}, and cardinality of 1 = cardinality of (Gj+ ~ {nG^U0)), and is a semi-value group of a semi-valuation of the field k. Example 111.14. Let C = Z, the group of integers with the usual * 4 * s , total order; nZ = {j f Z : j = n for a fixed integer n ^ 1} . Let P = nZ+ U {0} . Then, cardinality of (Z+~ P) = n - 1 . By Corollary 111.12 if A is the semi-value group of a semi-valuation of a fie ld k, U JKJ then [Al , © ( © (k,+»] © Zp is a group of divisibility. In particular, if k is a finite field GF(pm), then, as we have noted in Example 111.13, for q = {pm-I ) / { pn-1) where r divides m, [Z/(q) nr 1 _ ^ ( (+) 2/(p)] Q Zp is a group of divisibility for each integer n ^ 1 . 1 In Theorem 111.5, the submonoid P of Cw was assumed to be j , w( 1 (\ R ) Lf f 0} where w was the semi-val uat i on of the ring R = T(R) with semi-value group Cw, and I was a proper ideal of Rw. Thus, we were assured that w ^( P—0) generaged a proper ideal of R . Theorem w 111.15 below gives somewhat of a converse situation to Theorem I I I .5 . Theorem 111.15. Suppose that the sequence 0 -* A^ & ( Cw)p“* 0 is lexicographically exact, (PMD) + Cw ^ P , and v is a semi-val uat ion of a ring R = T(R) with semi-value group and semi-valuation ring R^ . Suppose that w =* /3v: By -> Cw is a semi-valuat ion of R with semi-value group Cw and semi-val uat ion ring Rw , Then, i) if Au 0, w '(PM)) generates a proper ideal I of Rw; I £ Rv £ . i i) Assume that Rw/I = T(Ry^/1) , that the canonical map h : Rw -» R^/* 43 maps U(R>w) onto U(RW/I), and that I £ the intersection of all regular maximal ideals of R . Then there exists a semi-valuation u of Rw/I w w having semi-value group and for which the commutative diagram D of Theorem I I I .5 is valid. Proof, i) Suppose that Xj , X 2 , . . . , x , x^+...+xn e R and w(x.)e P — 0, i ° 1 , . . . ,n . S ince P S Cw+ and w is a semi-valuat ion, w(x^+.. .+* n) = 0. Suppose that w (x j+ .. ,+xn) = 0 . Then v{xj + . . .+* n) e of(A^) . If A w 0, there exists a ^0 in A ; thus a' = or(a) + v(x. + +x ) e Q?(A ) . u u 1 n u However, v(x.) = a' , since /3(v(xj) -a1) = £v(x.) » w(x.) e P ~ 0 implies v(x.) - a1 - 0 by the lexicographic ordering. Since v is a semi valuation, we therefore have v(xj+...+xn) ^ a1, or, v(xj+.. .+xn) -a1 = - Qf(a) = 0, contrary to the assumption that a ^ 0 . Thus w(xj + . . .+xn)> 0. Since R is a semi-valuation ring, by Definition 11.8 t Rw consists of w m -1 + » 0 and all elements x e £ w (C ) for each integer m^ 1 , Also, J W ' + *“ 1 + — 1 (P~0) + Cw ^ P implies that for any re w (C^ } x e w (P— 0), we have rx e w ' (P ~ 0). Then, sinee 1 is generated by w ^(P ~ 0) and m -I + > r € Rw impl ies r = 0 or r e £ w (Cw ) for some m = I , we have 1 _ j that R I I . Hence, we conclude that w (P ~ 0) generates a proper ideal of R . That I ^ R £ R follows from the lexicographic ordering w — v w of the sequence (cr, ft) and the definition of a semi-va1uation ring. ii) Let h denote the canonical homomorphism of R onto R / l . Let r w w h1 be the restrictio n of h to U(Rw); h1 : U(RW) U(R / l ) . Define the homomorphism 1 ’uh11 * of diagram D of Theorem 111,5 to be the map or ^ vi . Since v is a semi-val uat ion, then "uhm also satisfied the axioms for a semi-vaIuation. Now define the homomorphism u of (Rw/l ) onto Au by the equation "uh,M =» uh1 . u is well-defined since ker h1 - (1 + 1)/"I — U(R ) = ^er " uhlM . u is a semi-valuation since h1 preserves addition and "uhIM satisfies the axioms for a semi-valuation. CHAPTER IV V-HOMOMORPHI SMS AND LEXICOGRAPHICALLY EXACT SEQUENCES In the converse situation of the previous chapter (Theorem 111.15) we assumed that the sequence 0 -* A^ ® ()p -♦ 0 was lexicographically exact, that Bwas the semi-value group of a semi-valuation v of a ring R = T(R), and that /9was a semi-valuation of R = T(R) with semi-value group C . This is always the case if B : B -* C is a V-homomorphism. W V w Thus, we investigate necessary and sufficient conditions for the homomorphism j3 : -• Cw to be a V-homomorphism, where 0 -* A^ ^ By ^ (Cw)p -+ 0 is the sequence L of Diagram D (Theorem 111.5(a)). The following example shows that if0-*A^>B^Cp-+0 is lexicographically exact and £ : B -* Cp is a V-homomorphism, then B : B -» C is not necessarily a V-homomorphism. Example IV .1. Let Z be the ordered group of integers with the usual total order. Then P = {k e Z+ : k = 3 or k - 0} and P1 = (k e Z : k = 2 or k 5 0) are submonoids of Z+ with P G P' . The sequence Qt 3 0 -* 0 -* Zp -♦ Zp -• 0, where ft is the identity map, is clearly lexico graphically exact and jS is a V-homomorphism. However, Zp ^ Zp, is not a V-homomorphism : k = in f7 {3,5}, but k £ in f7 T3,5l = 3. t|, ^ Let P be a submonoid of C such that (P 0) + C £ P and assume that 0->A^*B^*Cp--*0 is lexicographically exact. To avoid ambiquity, whenever we include elements of P and of C+ in the same discussion, we shall write c = 0 for c e P and continue to write c - 0 for c e C+. P *+5 <+6 We consider the following properties: Pr 1: For bp b2 ( B, b < for all b < implies /?(b^) = fl(bj) . Pr 2; For Cp Cj f C, c ft whenever c ft implies . Lemma IV .2. Let P be a submonoid of C+ such that (P — 0) + C+S P a R and assume that 0-*A-*B^Cp-*0 is lexicographically exact. Then Pr 2 imp!ies Pr 1. Proof. Suppose that Pr 2 holds and assume that for bp b2 e B, b < b2 for all b < b^ . Then, for all b < bj with /3(b) < 0(bj) , we must have 0(b) p 0(b2) . Suppose that /3(b) = 0 (b2) * T^en ^ ^ 2) P" 0 (bj) implies b2 < bj by the lexicographic exactness of (o, 0) . But b < b2 for all b < b^ implies b2 < b2, a contradiction. Therefore, /3(b) < j8(b„) . P For any dl <: 0 (b j), choose a preimage d such that 8(d) = d1 . Then d < bj . Hence, /3(d) < 0(b2) . 0(d) = d1 0(b2) for all d' ft 0(b1) implies, by Pr 2, that 0(bj) = 0(b2) . q.e.d. The fo il owing example shows that the converse of Lemma IV .2 is false. Example IV .3. (Pr 1 does not imply Pr 2). Let Z be the ordered group of integers with the usual total order. Let P = {k f Z+ : k = 2n, n e Z+} , P' = fk e Z+ : k £ 3 or k = 0} , P" = (k f Z+ : k - 2 or k = 0} . P1 eZ P" . The following sequence is lexicographically exact, where of, 0 are the usual injection and projection maps, respectively: 0 - Zp - Zp © (ZpnVp, 5 (Zpll) pl - 0 . We show first that Pr 1 holds, that is, if (c, c1) < (m2, n2) for all hi (c, c 1) < (nij , r»j) , then /3(, n^) ^ /?(, n2)(w ith respect to P11) , Suppose that |3(mj ,n^) = nj =» n2 . Then r^-n^ = + I , or n2“n^ = -1 , But ( -2k n^) < (m^ , n^), k= 1,2,,.., implies (rtij- 2k n^) < (m2, n2) by our assumption that (c, c1) < (m2, n2) for all (c,c‘) < (nip n^) . Thus, (m2 + 2k-nip± I) > 0, a contradiction to the lexicographic ordering of (o', /?) . Therefore, n^ = /3(mp n^) ^ 8(m2, n2) = n2 , and thus Pr 1 is satisfied. On the other hand, k 4 for all k 3, but 3 h since \ i P" . Hence Pr 2 is not satisfied. q.e.d. However, we have the following. Lemma IV. 4 . lf0-*A^*B^Cp-*0 is lexicographically exact and A+ = 0, then Pr 1 is equivalent to Pr 2. Proof. By Lemma IV .2 we need only show that Pr 1 implies Pr 2 if A+ = 0. Suppose that Pr 1 does not imply Pr 2. Choose c j , c^ e C such that c* <: c^ for all c 1 cj , and cj £ c^ . Let Cj, c2 be preimages of cj , cJ,, respectively. Then, there exists c < c^ such that c ^ c2; otherwise, Pr 1 would imply that fl(cj) = cj _ j3(c2)= c^ , contrary to our assumption that cj ^ . For any such c, however, we must have 8(c) = jft(cj) because if ,fl(c) <: 3 (c j), then by our assumption that c l < c£ = ^or c ' ^ ci = ^ cl^' we *iave that /3(c) < ft(c^) , and hence c < c^ by the lexicographic ordering of (or, /3) . Therefore, Cj - c e ar(A) . c^ - c c <*(A) and c^ - c > 0 imply that Cj-c e «(A) /"I B = ar(A ) . Therefore, A £ 0. q.e.d. Lemma IV. h applies in particular whenever A = {0} or A is a torsion group. i»8 Lemma IV ,5. Suppose that 0-*A^*B^Cp-*0 is lexicographically exact and that (P — 0) +■ C+^ P. If Pr 2 is satisfied, then 0 : B -• C Is a V-homomorph ism. Proof. Let bQ ^ inf0 {bj , . .. .b^) . Suppose d' «■ fl(b j) ...... and choose a preimage d of d1 in B. For any c' e C such that c< ^ <*' - 0(d) > we have 0(c) £ 0(d) for any pre image c of c 1 . 0(bj - c) = 0(bj - d) + B(d-c) Therefore, b. - c > 0, by the lexicographic ordering of (a?, 0) . Since this is true for any preimage of c 1 , b. *= c+a for all a e <*(A) . b “= inf_ { b , , . . . , b ) implies b c+a for all a e cv(A) . o D l n o If i8(bo) = 0(c+a) = 0(c), then bQ - c - a e of(A) . Therefore, bQ ^ c + (bQ-c-a) for a ll a e Qf( A) , and hence 0 ^ -a for all a e or(A) . This implies that A = 0. Assume momentarily that A =■ 0 and 0(kQ) = /3(c) . Then bQ =c. Thus, 0(bQ) < /3(d) I f b, = bQ for some i, then 0(b.) = 0(bQ) ^ 0(d) , Suppose, then, that b. - b > o for all i = 1 ,... ,n . Then, for all i and each i o ’ j = l,...,n , (b. + d)+(bj -bQ)- 0 . This implies that for each j = 1, . . . ,n, b ® d - b. + b , and thus d ^ b. , d b. for each o j o ’ j j j = l , . . . , n and bQ =infg [ b j , . . . , b } imply that bQ = d . Hence 0(bQ) ~ 0(d). Thus, if A = 0 and 0(bQ) = 0(c) = c' for some c' < d* , then /? is a V-homomorphism. If A = 0 and 0(bQ) s4 0(c) for all 0(c) = c' < d1, then c1 ,= 0(c) < 0(bQ) for all c1 < dl = 0(d) implies, by Pr 2, that 0(d) ^_0(bo), i.e., Q is a V-homomorphism. *t9 Assume now that A 0, Then 0(bQ) j* 0(c) = c' for all c 1 < d1 , for we have seen that /?(t>0) = /3(c) = c' for some c' < d' leads to A = 0 . Thus, 0(bo) £ 0(c) . c' = fl(c) p 0(bQ) for all c' = fl(c) < 0(d) impl ies, by applying Pr 2 again, that 0(d) ^ 0(bQ) . Hence 0 is a V-homomorph i sm. I pmma IV . 6 . Let 0 : B -» C be any V-homomorphi sm. Then, if 0 is not an order isomorphism, 0 satisfies Pr 1. Proof. Suppose that c < c^, for all c < , where c, c^, € B . S ince Js not an order isomorphism, there exists b C B such that b ^ 0 and 0 (b) * 0 . Choose such a b . Let c = Cj, Cj+b . c = Cj implies b*^ 0, contrary to the choice of b. Therefore, c < C| . By our in itia l assumption, c < C 2 . Therefore, c 2 - infg { c j , c^+b} . Since 0 is a V-homomorphism, 0 (c 2) ^ »nfc [0 ( ^ 3 , 0 ( 0 ^ ) } = 0 (c ,) . Theorem IV .7. Suppose that 0-+A^*B^Cp-»0 is lexicographically “f* exact and that (P ~ 0) + C S. P . If A is filte re d and 0 is not an order isomorphism, then 0 : B -* C is a V-homomorphism if and only if satisfies Pr I: For Cj, c 2 € B, c < C 2 for a ll c /J(e,)* 0(cj) . P ro o f. If 0 is a V-homomorphism and not an order isomorphism, then £ always satisfies Pr 1, by Lemma IV . 6 . Hence, suppose that A is filtered and 0 satisfies Pr 1. Let bQ = infg {bj,...,bn3 . Suppose that d 1 ^ 0(bj) ,...,0(bn) , and choose a preimage d of d 1 in B. Suppose c < d. Then 50 0 (c) ■= /3(d) = 0 (b.) , i = I ...... n, impl ies 0 (c) ^ 0 (b.) or 0 (c) = 0 (l>;) , by the property (P ~ 0) + C+£ P . Thus b. ■ c > 0 or b. - c e cv (A) , the first assertion following from the lexicographic ordering of (or, Pt) . In either case, c ^ b. - a. for some a. € or(A) . Since A is filtered, there exists a e or(A) such that a = . There fore, c ™ b. - a, and hence c+a * b , , . . . ,b . Thus, b = * i 1 n ' o inf0 {b.,...,b } implies that b = c + a. D i n o If b - a = c, then b - a ^ b. -a . for all i impliesb ^ b. + o * o I I o I a - a. £ b., since a = a. . If 0(b ) = 0(b.) for some i, then i i ’ i o I 0 (bQ) = 0 (c) 0 (d) ^ 0 (b.) implies 0 (bQ) = 0 (d) . Thus, suppose that 0(bQ) ^ 0(b.) for all i. Then, bQ ^ b . implies 0(bQ) 0(b.) and hence bQ < b. for all T . Since (P ~ 0) + C+^ P 0, we have that 0 (b. - d) + 0 (b. - b ) f P~ 0 for all i and for each i j o j = I _,,.,n. Therefore, b. - d + b. - b - 0 for al1 i and each J ’ ’ i j o j = ),...,n. Hence it follows, by applying bQ ^ inf^ [b j,...,b n] twice, that bQ ^ d and thus 0 (bQ) ^ 0 (d) , We have thus shown that whenever c < d we always have c ^ bQ - a, a e a(A) , and if c = b - a for some c < d, then 0 is a V-homomorphism. o It thus remains to consider the case that c / b - a for all c < d. o In that case, c < b - a for all c < d . By Pr I, d^ b - a and ’ o o hence 0 (d) 0 (bQ - a) = ^(^D) • rv O Theorem IV .8 . Suppose that 0 -* A — B ^ Cp -* 0 is lexicographically exact and that (P — 0) + C+£ P. If A is not filte re d , then 0 is a V-homomorph i sm if and only if C satisfies Pr 2: For cj , c^, € C, c' < cjj for all c 1 cj implies cj ^ cJ, . 51 Proof. Suppose that A is not filtered, and that C satisfies Pr 2. Let bQ = infg £b] , . . . ,bn} . Suppose d' = /3(bj) , . . . ,/9(bn) , and that c 1 <£ d‘ . For any c e B such that j5(c) = c' , fl(b.) - d1 ^ c' v 4* ~>- implies b. = c, by applying (P ~ 0) + C £ P ~ 0 . Therefore, bQ = c . Since this is true for any preimage of c 1, we conclude that b - c+a for any a c Qf(A) I f b '= c, then 0 = a for any a e a (A ), which would imply that A = 0, contrary to our assumption that A is not filte re d . Thus, bQ > c for any c e B such that jS(c) = c 1 . I f j3(bQ - c) = 0 , then bQ - c e <*(A) and thus + bQ - c) = 8 (c) = c'. Therefore, b >c+b - c, a contradiction. Hence, o o 0(bo) ^ 8(c) = c' , since bQ > c and #(bQ) > /3(c) = c 1 . Therefore, Pr 2 implies /?(bQ) = d' . Conversely, assume that A is not filtered, and that ^ is a V-homomorphism. Let c j , c^ be elements of C such that c 1 ^ c^ for a ll c' ■j* cj . Choose a preimage C| of cj in B. Since A is not filtered, we can find a^ , a^t e o(A) such that there does not exist a e or (A) with a ^ a^ , a^ . Suppose that d ^ dj + a^, Cj + , Then £(d) = cj . I f S(d) = c j , then d = c^+a for some a e ff(A), and then a ^ a1, a2, a contradiction. Thus, g(d) ^ cj . Hence, £(*0 ^ c2 * Therefore for any preimage C 2 e B of c!>, C2 = d . By the choice of d and the assumption that 8 is a V-homomorphism, we conclude that c^ - in fc fjSfc^+a^), fl(cj+a2)} = cj . Corol1ary IV.9. Suppose that C is a lattice-ordered group and P is a submonoid of C+, (P — 0) +C+ £=P, such that P satisfies. Pr 3: There exist , kj f P such that kj ^ kj and k 2 ^ k | , and d = Infq (kj, k2] implies d = kj, k2 . Then 8 is a V-homomorphism. 52 Proof. It suffices to show that Pr 2 is satisfied. Suppose that c' < for a ll c 1 cj . Then cj - k ^ cj for a ll k 0 implies cl - k < cl for all k > 0. Let k. , k, f P such that k., k, sat i sfy I p z p 1 / i *■ Pr 3. Let c = inf^ fc£ + k p c^ + k^} . Then c j= cjj + k p c^ + k2 . Therefore, c < c^ + kp c^ + k 2 ? since k| k2 , kj ^ kj . Hence, c^ + kj - c, - c ^ 0 . Thus, cj jj cl, + (c^+kj-c) , c^-ffc^+k2“c) implies cj-c^+c jy c£ + k p c^ + k2 . Therefore, c ^ infj. fc^ + k p cl, + k23 impl ies c = cj - cl, + c , or = cj . Therefore C satisfies Pr 2. q.e.d. We note that if C is la ttic e , then C+ satisfies Pr 3 t r iv ia lly . CHAPTER V A CLASS OF GROUPS WHICH ARE NOT GROUPS OF DIVISIBILITY OF DOMAINS In [5 ], Jaffard provides an example of a filte re d group which is not a group of divisibility of a domain. Ohm constructs a large class of such groups of the form A (+) C where A £ {0} and C belongs to a certain class of lattice-ordered groups. In this chapter we enlarge this class of groups to include those of the form A (+) Cp where P is a submonoid of C+ satisfying a specified property. Finally, we show that an unpublished result of R. L. Pendleton - that the only filte re d orders on the group of integers Z which produce groups of d iv is ib ility of domains are the two obtained by taking as positive elements either Z or -Z - follows from a more general proposition. Lemma V. 1. Suppose that 0-*A^B$Cp-,0 is lexicographical ly exact and v is a map of T(R)" onto B. Let w - /?v : T(R) -* B -* C . if A t* 0, and if there exist x, y, x+y e T(R) such that w(x+y) w(x) , w (y), then v is not a semi-valuation. ([1 3 ], Lemma 5 .1 ). Proof. w(x+y) <: w(x) , w(y) implies that v(x+y) + a ^ v(x) , v(y) for all a ; a(A) . Therefore, if v is a semi-valuation, we must have v(x+y) = v(x+y) + a, for all a e cy( A). This implies A = 0, a contradiction. q.e.d. Let C be a lattice-ordered group and let i be a V-imbedding of C in the ordered direct produce D = II D of totally ordered groups u u (See [5] for the existence of such an i.) . Let py be the projection 53 5^ of D onto D and let i = p 1 . By Jaffard's Theorem [5] , C is the u u r u ' semi-value group of a semi-valuation w of a field K (and hence of a r i n g R = T(R) which is n o t a f i e l d : If X is an indeterminate over K, we simply take R = K[X] @ M, the ring formed by the principle of idealization, with M = Q (K[X]/B.) where {B.} = set of proper ideals i of K[X] . Then, if D is the semi-valuation ring of a semi-valuation w or K with semi-value group C, D (+) M is a semi-valuation ring of a semi-valuatIon w 1 of K[X] (+) M with semi-value group R’/U(D (+) M) = iz K /U ( D) rj C.) . Since the projection maps p^ are always V-homomorphisms whenever the 0 are filtered, u ~ i w Is also a semi-valuation for each u ’ u u. Moreover, i = ir i and hence iw = iru . u u 4* Let P be a submonoid of the positive elements C of the la ttic e - ordered group C, We say that Cp satisfies property ©rif there exist Cj, c2 e C such that Cj ^ c 2 and c 2 ¥" C j, and such thati u(c ^ ^ *u(c 2) for al I u, and d = infj,[cj ,c2) impl ies d ^ c^ ,c 2 . Any ordered direct product C of at least two copies of Z satisfies property & for some submonoid P of C . For example, if P = C ; or, for c^, c2 e C such that c^ ^ c 2 and c 2 ^ Cp and i u (c ^) £ i o^) for all u, let d= inf^fcpC^ ; let a be any minimal element of C — 0 such that a j* C|-d, c2~d . Then P = C ~ (a} is a submonoid of C+ : Ifx, y f P, x^O and x+y = a, then x = x+y = a implies 4* that x = a by the minimality of a in C ~ 0, contradicting x C P . Hence x+y e P . If Cj-d and c2~d are the only minimal elements of C ~ 0, then C p c 2 + c2 satisfy the first two conditions of property and thus, if d 1 = infj,{cp c2+c2) , we can take P = C+ ~ £c 2-d'3 . 55 In either case, it is evident that Cp satisfies property©; and P ^ C since a e C , a 4 P . Lemma V .2. Let w be a semi-valuation of a ring R = T(R) with semi- value group C . If C is to ta lly ordered, and Xj , x2 , xj +x2 ? ^ ' then w(xj) £ wtx 2^ implies w(xj+x2) = inf^fwfxj), w(x2)} . Proof. Since w is a semi-valuation, w(xj+x2) = in fc {w(x^), w(x2)} . Since C is to ta lly ordered, we may assume w(xj) < w(x2) , since w(x|) i4 w(x2) by hypothesis. If w(xj+x2) > w(Xj)= w((xj+x2) - x2) , then w(xj) = inf (w(xj+x2) , w(x2)} >w (xj), a contradiction. Therefore, w(x^+x2) = w(x^) = inf^fw(x^), w(x2)} . Lemma V.3. Suppose that C is a lattice-ordered group and i is a V-imbedding of C in the ordered direct product 0 = ID^ of to tally ordered groups, and assume that w is a semi-valuation of a ring R = T(R) with semi-value group C. If C satisfies property©; then there exist Xj , x 2 e T(R)“ such that whenever xj +x2 f T(R)** , then w(xj+x2) = infc{w(xj), w Proof. Choose x^, x 2 € T(R)' such that w(xj) = , w(* 2^ = c2 * Let u = i^w, where, as we noted in the discussionfollowing Lemma V .l, iu = pui; p^, the projection map of D = IDu onto . u(x^) # u(x 2^ implies u(xj+x2) = infg (u(x^), u(x2)} . Since iw = iru, w(xj+x2) = u inf^fwfxj), w(xj)J . By hypothesis, Cj and c 2 are unrelated, so infcfw(x,), w(x2)} Theorem V.*t. Let C be a lattice-ordered group and P a submonoid of C such that (P ~ 0) + C £ P and Cp satisfies property O, Suppose that 0-*A^B^Cp-*0 is lexicographically exact and A ^ 0 . Let v 56 be a map from the group of units R* of a ring R = T(R) onto B. If Cj ■= flv(xj), Cj =* /?v(k 2) satisfy the conditions of proper(y £■)'and Xj+x^ € R , then v is not a semi-valuation. ([13], Theorem 5.1) Proof. Suppose that v is a semi-valuation. By Corollary IV.9, /3 : B-* C is a V-homomorph i sm. Therefore, w = fiv is a semi-val uat ion with semi-value group C. I f Xj+x^ € T(R) , then by Lemma V.3, w(Xj+x2) = in fc{w(Xj), w(x2)} < w(Xj), w(x2) . Since Cp satisfies property©; w(Xj+x2) ^ w (Xj), w(w2) . Thus, by Lemma V .l, v is not a semi-valuation. Corol1ary V .5 . Let A be any non-zero ordered group, and let P be a submonoid of a lattice-ordered group such that (P ~ 0) + C+ S: P ^ C+ and such that CD satisfies property (9*. Then A (+) C is not a group r P of divisibility of a domain. Proof. A / 0 and 0 A A (+) Cp -5 Cp -* 0 is lexicographical ly exact, with i and p the usual injection and projection maps, respectively. By hypothesis, Cp satisfies property O'. I f D is a domain with JL ^ quotient fie ld T(D) and v is a map from T( D) onto A (+) Cp, and if Cj = pv(Xj), c2 = pv(x2) satisfy the conditions of property then x 1+x 2 c T(D)" , since T(D) is a fie ld and Cj and c 2 are unrelated. Thus the conditions of Theorem V.5 are satisfied. Therefore, v is not a semi-valuation. Thnce, A @ Cp is not a group of d iv is ib ility of a domain, if A ^ 0. Corol1ary V. 6 . Let P be a submonoid of a lattice-ordered group C such that ( P ~ 0) + C P , and assume that C = IT>u the ordered 57 direct product of to ta lly ordered groups Du. If P contains at least two s tric tly positive disjoint elements (in C ),Cj, c2, such that ^ Pu^C2^ ^or u’ anC* ^ ^ thenA (+) Cp is not a group of divisibility of a domain. Proof. Suppose that c^ , c 2 are disjoint elements of C such that c^, c2 e P ~ 0 . Then c^ ¥ c2 and c^ ^ c^ . Si nee c^ and c 2 are disjoint, 0 = infj.{cpC2} . By hypothesis, p^fcj) ^ pu(c2) for all u, and Cp c 2 e P . Thus, 0 ^ c] »c2 * Therefore, Cp satisfies property By Corollary V.5, if A £ 0, then A (+) Cp is not a group of divisibility of a domain. q.e.d. Thus, for example, if A / 0, and Pisa submonoid of Z x Z, the ordered direct product of two copies of the integers, such that (P — 0) + (Z x Z) + ^ P £= (Z x Z) + , then A @ (Z x Z) p is never a group of d iv is ib ility of a domain if P contains both (0, m) and 4* (n, 0) for some m, n e Z — 0 . We have already shown that Zp, P ={j : j = n for some n > 1 UtO) is never a group of d iv is ib ility of a domain (Example 111.7). This fact also follows from the proposition below. Proposition V .7. Let C be an ordered group and let P be a proper submonoid of C . Let i : Cp C be the identity homomorphism. Let v be a map from the group of units R" of a ring R = T(R) onto C„, and assume that iv is an additive semi“731uation with semi- P V* valuation ring D and semi-valuation monoid D . Assume also that for each regular non-unit x of D, there exists t C U(D) such that x+t e D*', Then v is not a semi-valuation with semi-val uat ion ring D‘ and semi-valuation monoid O'" . Proof. Let x e R* such that lv(x) > 0 . Let t e U(D) such that x+t € D" . By Proposition 11,16, since iv is additive, the regular non-units of D generate a proper ideal of D. Therefore, x+t € U(D) , Since iv is additive, iv(x) > iv(t) = 0 implies iv(x+t) = 0. Since i is an injection, this implies v(x+t) = 0 . Hence, if v is a semi valuation with semi-valuation ring D1, we must have x+t, t € D‘ . But then, x+l e D' , - t € D' imply x e Dl . Hence D and D' have the same regular non-units. But this implies P = C+ (CJ(D) = U(D1) because I is an injection), contrary to our assumption that P is a * 4 * proper submonoid of C , Therefore, v is not a semi-valuation with j . semi-valuation monoid D1" . q.e.d. Proposition V.7 applies to the following situation: Let C be a * 4 * to ta lly ordered group and let P be a proper submonoid of C such that Cp is f i 1tered and (P 0) + C+£ P . From Lemmas !V.*t and IV .5, the identity map i : Cp -» C is a V-homomorphism. Hence, if v is a semi- valuation with semi-value group Cp then iv is an additive semi valuation. Thus, from Proposition V.7, v cannot be a semi-valuation of a domain (in the case of a domain, the ring D of iv is quasi local and thus the condition that there exist t e U( D) for every non-unit x ^ 0 such that x+t € D is automatically satisfied.) . Since the composition of two V-homomorphisms is again a V-homomorphism, Proposition V.7 also applies, in the above situation, to any submonoid P1 P such that the identity homomorphism i : Cp, -• Cp is a V-homomorphIsm. We consider now the submonoids P £ Z + such that Zp is filte re d . *1* Claim I . Let P ^ Z such that Zp is filte re d . For each integer j € Z+ - 0 and for all i = 0 ,1 ,...,2j, there exists k eZ+ such that j . 2k + i e P , Proof. Since Zp is filte re d , 1 = m-n for some m, n e P. Hence P contains two consecutive integers k, k+1 . But then 2k, 2k+l, 2k+2 also belong to P since P+P£ P . Assume inductively that j-2k+i e P for some j € Z ~ 0 and for a ll I = 0,1 ,.. . , 2j . (j + 1). 2k+i = (j ■2k+i)+ 2k . I f i = 2j , then j*2k+l e P by the inductive hypothesis. 2k e P by the j = 1 case. Therefore, their sum is in P. If i = 2j+l, then (j+l) . 2k+(2j+l) = (j-2k+2j) + (2k+l) f P + P, again by the inductive hypothesis and the j = 1 case. Sim ilarly, for i = 2j + 2 , ( j . 2k+2j) + ( 2k+2) c P + P £ P . Hence, (j+1) • 2k+i C P for each i = 0 , 1 , . . . ,2{j + l) . -f- Claim 2. P contains every integer n such that n-k*2k e Z Proof. By Claim 1, k-2k+i e P for all i = 0 , 1 , . . . , 2k . Then (k*2k+2k) + 1 = (k+1)k+1 e P, also by Claim 1. Suppose inductively that (k*2k+2k) + r e P for all r = n, r > 1 for some integer n. If n < 2k, then k-2k + 2k + n = (k+l)k+n c P by Claim 1, so we may assume that n = 2k and write n = s.2k+t, t < n . Then ( k- 2k+2k) + (n+l) = (k*2k+2k+t) + ( s-2k+l) . k-2k+2k+t e P by the inductive hypothesis, since t < n, and s*2k+l f P by Claim 1, since s = 1 . Therefore, the sum (k-2k+2k) + (n+l) (k*2k+2k+t) + (s-2k+l) is in P. 60 Proposj t ion V .8 . Let A, B be ordered groups, and let h : A -* B be a surjective order homomorphism. Assume that B is totally ordered. If the set B+ — h(A+) has a maximal element, then h is a V-homomorphism. Proof. Let a « inf. fa.,...,a } . Since B is totally ordered, we o m i n may assume that h(a^) = h(a2) ^ ...= h(ap) and it thus suffices to prove that h(aQ) = h(aj) for some j = l,...,n . Suppose that h(aQ) < ^ (aj) f ° r each j = l , . . . , n . Let d 1 be a maximal element of B+ — h(A+) . Since h is surjective, we can choose a pre image d of d 1 in A such that h(d) = d1, d £ A+ . h(aj-aQ) > 0 for all j = 1 n . Therefore, h(a^.) - h(aQ) + h(d) > h (d ), 0. Since h(d) is maximal in B+ h(A+) , this imp]ies that a.-a+d e A+ . a. = a -d for all j=1,,..,n implies that j o j o a = a -d . Hence 0 --d , contrary to the choice of d. Thus, 0 o ’ ' h(aQ) = h(a^) for some j =l,...,n. q.e.d. By Claim 2, if Z+ / P, Z+ ~ P is fin ite and hence Z+ ~ P has a maximal element. Thus, by Proposition V . 8 , the identity map 1 : Zp -* Z is a V-homomorphism. I t follows, therefore, from Proposition V.7, that Zp is not a semi-value group of a domain for any P ^ Z + such that Zp is filtered. Since any ordering on Z is contained in either •j* Z or -Z , this implies that the only filtered orders on Z which produce *(■ groups of d iv is ib ility of domains are Z or -Z BIBLIOGRAPHY 1. N. Bourbaki, A1 gebre. chapitre 6 , "Groupes et corps ordomu’s" (Hermann, Paris, 1964), 2. N. Bourbaki, A1gebre commutative, chapitre 6 , "Valuations" (Hermann, Paris, 1964). 3. N, Bourbaki, Algebre commutative, chapitre 7, "Diviseurs" (Hermann, Paris, 1965). 4. W. Heinzer , J -Noether ian integral domains wi th _1_ _m the stable range. Proc. Amer. Math. Soc. 19 (1968), 1369-1372. 5. P. Jaffard, Les systemes d'ideaux (Dunod, Paris, I960). 6 . I. Kaplansky, In fin ite Abelian Groups, (Univ. of Michigan Press, Ann Arbor, i960). 7. W. K ru ll, Allgetneine Bewertungstheor?e. J. Reine Angew. Math. 167 (1931), 160-196. 8 . W. Krull, i dealtheorie in Ringen ohne Endlichkeitsbedingung. Math. Ann. 101 (1929), 729-744. 9. S. Lang, A1qebra. (Addison-Wesley, Reading, 1965). 10. J. Marot, Jne generalisation de la notion d'anneau de valuation. C. R. Acad. Sc. Paris, (1969), 1451-1454. 1!. M. Nagata, Local Rings. (1 nterscience, New York, 1962). 12. T. Nakayama, Ori Krul 1 1 s con jecture concerning completely integral ly closed integrity domains I, I I , Proc. Imp. Acad. Tokyo 18 (1942), 185-187; 233-236; 111, Proc. Japan Acad. 22 (1946), 249-250. 61 62 13. J. Ohm, Semi- valuations and groups of d iv is ib ilit y . The Canadian Journal of Math. 21 (1969), 576-591. 14. G. Scheja and U. Storch, Bemerkungen Liber die Einheitenqruppen semilokaler R i nge Mathematisch Phys i kali sche Semesterberichte , XVII (1970), 168-181. 15. D. Underwood, On some uniqueness quest ions in pr imary representat ions of ideals. Journal of Math, of Kyota Univ. 9 (1969),69-94. 16. 0. Zariski and P. Samuel, Commutative Algebra 1, II (Van Nostrand, New York, 1958, 1961). APPENDIX A .I. Let R be a quasi-local ring of the form k+M, where k is a field and M is the maximal ideal of R. Then ( 1) the mapping (a+m)(l+Mn) ^ (a,(l+a *m)-(l+Mn)) is a group isomorphism from (R/Mn) ' onto k" x (l+M )/(l+hn) ; (2) i f D is a subring of k, the mapping <«.’ induced by (R/Mn)'7u(D) ^ k*7u(D) x (l+M)/(l+Mn) , is an order isomorphism, with (R/Mn) 7U(D) ordered by D‘7 u (D) and k"/U(D) x (1+M)/(1+Mn) ordered by (D */U (D ),1) ; ( 3) (l+M )/(l+M n) and M/Mn are isomorphic as groups. Proof. (l) Cg is wel 1-def ined: Suppose that (a+m). (l+Mn) = (a'+m1). (1 +Kn) . Then (a+m) (a'+m1) € 1+Mn . Let (a ’+m ')"1 = a" + m", a1, a" e k, m 1 , m" e H . Then (a+m) (al 4-m") = aa" + a"m+am" + mm" € 1+Mn implies aa1 = 1 . ( l+a 'm )(l+a‘ ^m1) 1 = a '(a+mja 1 ^(a'+m1) 1 = a *a‘ (a+m) (a'+m1) . (a'+m 1)"1 = l/(a'+m') = a" + rtf' implies 1 = a 'a " + a'm" + a'^'+m'm" e M, and l-a'a" € k . Therefore, 1 - a 1 a" = 0, and thus a 1a" = 1. Hence aa" = 1= a'a" implies a = a 1 . By assumption, (a+m)(a'+m1) e 1 + Mn , _ * _ | _| p and we therefore conclude that (l+a m )(l+a‘ m') e 1+M . Thus ( l+a "1 m) • ( 1+Mn) = (l+a'^m'Ml+M0) , Q is a homomorphism: CQ[(a+m) ( 1+Mn) ] CC [(a'+m 1) (l+Mn) ] = (a ,(l+ a 'm) ( 1+Mn) ) (a 1 , ( I+a' ' m1) (1+-M ) ) = (aa',(l+a ^m)(l+a' ^ m') (1 +Mn) ) =dC[(aa'+am 1 + a 1 m+mrn') ( 1+Mn) ] = 02 [(a+m) (a l+m') ( 1+Mn) ] = = Q?[(a+m)(l+Mn). (a '+ m ')( 1+Mn) ] . 63 6*t G2 i s b i ject ive: Suppose that C£[(a+m) ( 1+Mn) ] = ( I , l'(1+Mn)) . Then a = I; l+a = 1 implies a 'm = 0 . Hence m = 0 . Let (a, ( 1+m) ( 1+Mn) ) <■ k* x (l+M/(1+Mn) . Then, Cfl [(a+am)( 1+Mn) ] = (a, (l+a 1 (am) ( I+h") ) = (a , ( l+m) ( t+Mn) ) (2) LetCflbe the mapping in (1 ), p, p 1 the canonical homomorphisms, i the identity homomorphism in the diagram below. (R/Mn) * ------— ------k* x (l+M )/(l+M n) 1 P . erf * ' l (P' ,i' (R/Hn)“/U(D) - ■ k /U(D) x (l+M)/(l+Mn) ker (p 1 . iK lS ker p: p((a+m) ( 1+Mn) ) = 1 implies a e U(D), m e Mn . (p 1 , i) C£2 ((a+m) (1 +Mn) ) = (p1 , l)[(a, (l+a ]m) ( 1+Mn) ] = (1 , 1) . Hence CG‘ is defined canonically and is surjective since (p1, i)CC is surjective. If (p 1 , i ) 0? ( (a+m) ( 1+Mn) ) = (a, (l+a 1m)(l+Mn))= (1, 1), then a e U( D) , m e Mn ; hence p((a+m) ( 1+Mn) ) = 1. Thus, ker (p 1 , i ) d = ker p and C£*is therefore injective. I f x f D"/U(D) , then x = p((a+m) ( 1+Mn) ) for some a e D , m e Mn , Cfl.' (x) = Cfi'p((a+m) ( 1+Mn)) = (P1 , i) ( (a+m) ( 1+Mn) ) = ( P1 , i)[(a, (l+a 'm)(l+Mn))] = (a, l), where a e D" . Conversely, if (a, 1) e k* /U(D) x ( 1+M)/ ( 1+Mn) , a e D*. thenc£r , ( i , I) = p( (a+m) ( 1+Mn) ) where m e Mn . Hence i s an order isomorphism. 2 3 3 3 (3) M /M is a subspace of the vector space M/M over k(M/M becomes 3 3 a k-vector space by defining a(m+M ) to be the coset am+M .). Let a, /3 be the canonical injection and projection maps, respect ively: 0 - M2/M3 “ M/M3 ^ (M/M3)/(M 2/M3) - 0 . 3 2 3 2 Then (M /tT)/(M /MJ) pa M/M as vector spaces and there exists a 3 2 subspace W of M/M such that W ta M/M as vector spaces over k ([9 ], 2 Theorem 4, page 87; and page 76). Thus W and M/M are isomorphic as additive groups. By {[1 4 ], Satz 3 .2 ), the mapping i}) : a+Mn - (l+a)(l+M n) is a group isomorphism from M° '/Mn onto (1+Mn *)/(l+ M n) . Let tf,: M/M 2 - ( 1+M)/ { 1+M2) , (a+M2) = (l+a) ( 1+M2) , a e M ; 2: MZ/M3 - ( 1+M2) / ( 1+H3) , ii)2 (a+M3) = ( l+a) (1+M3) , a e M 2 . Then, a, 8 induce a 1, /3' such that 1 — ( 1+M 2) / ( 1+M3)0- (1+M)/(1+M3) (1+M)/(1+M2) — I is exact . Since or, $ are also group homomorphi sms, is a group isomorphism. Define $ : (l+M)/(1+M3) -M/M 3 by $ [(1+m) (1+M3) ] = (S |w) _1 [ ( 1+m) (1+M3) ] , i f m t M2 ; $ [ ( 1+m) ( 1+M3) ] = a 1 ^ [( 1+m) ( 1+M3) ], i f m e M2 . 3 2 3 Since [rrrt-M : m ^ M } is a subspace W of M/M isomorphic to 2 M/M , $ is well-defined, and is a group homomorphism since it is 2 defined as the composition of homomorphisms. I f m f M , then § is clearly injective because a, tfi2' , o'1 ^ are injections. Suppose that m £ M2 . $[( l+m) ( 1+M3) ] = 0 implies /?’ [ ( 1+m) ( 1+M3) ] = 0 , since I I 3 The assertion now follows from induction by assuming (inductively) that M/Mn ^ ( l+M)/( 1+Mn) , where $[( 1+m) ( ]+Mn) ] = ($ |w) ft1 [ ( 1+m)( 1+Mn) ] , where m i Mn ; $ [ ( 1+m) { 1+Mn) ] = * [ ( 1+m) ( 1+M3) ] , where m e Mn ; and from the diagram I > ( 1+Mn) / ( 1+Mn+1)— K 1+MiV(1+Mn+1) ^ (l+ K K l+ M n) - 1 Valuations of fields If w is a semi-valuation of a field K onto a totally ordered Abelian group G, then w is called a valuation of the fie ld K and G is called the value group of w. The ring of w is then a quasi-local domain with K as its quotient field, A.2 Let R be a quasi-local domain. The following are equivalent, (1) R is a discrete rank one valuation ring; that is, the group of d iv is ib ility of R is order isomorphic to the to ta lly ordered group of integers, Z. (2) R is Noetherian with a principal maximal idea! M; every ideal A of R, (A 0, R) is of the form Mn, n = 1. (3) R is a principal ideal ring. ([2 ], Proposition 9, page 109). A,3. (Krull*s Theorem). Let C be any to ta lly ordered group and k any field. Then there exists a valuation w of the quotient field of the group algebra Cl^(c) with value group C and ring R = k+M, where M is the maximal ideal of R. ( [ 2 ] ,, Example 6 , page 107). A.4. Let L be an extension fie ld of fin ite degree of a fie ld K, Let v be a discrete rank one valuation of K. Then, if v* is a valuation of L extending v, v* is also a discrete rank one valuation. ([2], Corollaire 3, page 140). 67 A.5. Let K be a field, v a valuation of K with ring R, and M the maximal ideal of R. Let L be a fin ite separable extension of degree n over K, and let B be the integral closure of R in L. Let C, C1 be the value groups of v and a valuation v1 of L extending v, respectively. Let e = index of C in C , f = dimension of B/mB as a vector space over R/M, g = number of maximal ideals of B lying over M. Then ef g = n, with equality holding if B is a finite R-module. ([2], Proposition 8, pages 152-153). A.6. Let R be a valuation ring of the form k+M where k is a fie ld and M is the maximal ideal of R. Let C be the group of d iv is ib ility of R (up to order isomorphism). Then the group (l+M )/(l+M n) is isomorphic to the additive group @ (k ,+ ), where (k,+) denotes the additive 1 group of the field k, andcardinality of 1 = the cardinality of [C+ ~ (nC+ U {0}) ], nC+ =fceC: c = C j+ .. .+cn> c j > 0, i = 1 , . . . ,n} . Proof. For each n » 1,2 Mn is an R-module. Mn -*/Mn, I = j < n , becomes an R-module by defining a(x+Mn) to be the coset ax+Mn for a e R, x € Mn -* . Then, since k £ R, Mn J*/Mn is also a k-module, and thus has a vector space basis over k. Let m^ = m + Mn) n _ | be the set of basis elements of M /M Let w be the valuation of T(R) with value group C. Define a mapping f : (3 -* w(Mn 1 — 0) ~ w(Mn ~ 0) by f(irr) = w(ma) . f i s wel 1-def ined: Suppose that y c C (3 * Then y-ni^ € Mn . w(y-m ) = inf tw(y ) , w(m )] . If w(y) w(m ) , then, since w is a O t O f O f valuation, w(y-m ) = infr {w(y), w(m )} . w(y-m ) # w(m ) since Oi w Q f O f Q? 68 m^ t Mn and y-m^ c Mn . wfy-m^) “ w(y) impl ies y c Mn, and then, since y-m e Mn, so does -m , a contradiction to m / 0 . Hence w(y-m ) # a cy ot a inf[w(y), w(ma)}, and therefore, w(y) = w(m^) , f is injective: Suppose that f ( ma) = ^(m^) , or w(raff) = w(m^g) ■ Then m = tmflI where t e U(R) . Since R = k+M. we can write t = a+m O' p where a e k, m e M. Then m = am_ + mm„ . m_ e Mn * and m e M imply ' Qf P B B mm^ e Mn , Hence = am^ = am^ . Since m^ c ^ , they are linearly independent over k. Thus a = 1. Hence tn^ “ m^ . f is surjective: Let d = w(y) e w(Mn * ~ 0) ~ w(Mn ~ 0) . Then y C Mn * ~ Mn . Hence y = y+Mn c M/Mn, y ^ 0 . We can write y as a linear combination of elements o f @ with coefficients in k: y = a.m +, , .+a.m .mj^m, , . „ _ _ 7 l a , JL a . ’ a. a. for i jt j . Then, y - a.m I & 1 J 1 Oj X a ^ = m € Mn , S ince w is a valuation, y-a.m -...-affm -m= 0 implies 1 a ] &CtZ (i) w(y) = w(a. ) = w(ma ) ^or some 1 * i i or (ii) w(a.m ) =w(a.m ) for some j i , ' i a. j a / J ’ • J J or ( i i i) w(y) = w(m), or (iv) w(a.m ) = w(m) for some i . i a. (ii) implies m = m , j t4 i , contrary to assumption. ( 11 i) “ i 01} implies y c Mn, a contradiction to y ^ 0 . (iv) implies m^ c M , a i contradiction to m e $> • Therefore, w(y) = w(m ) . Hence i i d = w(mi ), where m < 0 . Thus f is surjective. w » Of * I 1 — n _i Since f is b ijec tiv e, cardinality Qj = cardinality [w(M ~ 0) ~ w(Mn ~ 0) ] . M2/M3 is a subspace of M/M3 and dim^M/M3) = dim^(M2/M3) + dim^t(M/M3)/(M 2/M3)) ([9 ], Theorem 4, page 87). (m/ m3) / ( m2/ m3) « M/M2 as vector spaces over k ([9 ], page 76) . Therefore, dim^M/M3) = dim^(M2/M3) + dim^(M^M2) . S i nee [w(M —■ 0) ~ w(M3 ~ 0) 3 \J [w(M ~ 0) ~ w (H '■'•'0)3 “ w(H-^O) ~ w(M3 ~ 0) , and [w(M ~ 0) ~ w(M3 ~ 0) 3 f \ [w(M ~ 0) ~ w(M ~ 0) 3 = $ , •j the empty set, we have that card in ality [w(M ~ 0) ^ w(M ~ 0) 3 “ 2 2 cardinal i ty [w(M ~ 0) ~ w(M ^ 0) ] + card inal i ty [w(M ~ 0) ~ w(M3 ~ 0) 3 . Hence dim^(M/M3)= card in ality [w(M ~ 0) ~ w(M3 ~ 0) 3 . Assume inductively that dim^(M/Mn ') = card[w(M) ~ w(Mn *)3 . n j Let p : M/M -* M/M be the canonical k-module map. Then, dim^(M/Mn) = dim^ker p) + dim^(lm p) ([93, Theorem 4, page 87). Thus, dim^(M/Mn) = cardinal Ity [w(Mn ^ ~ 0) ~ w(Mn~ 0)3 + cardinal i ty n _i [w(M ~ 0) ~ w(M ~ 0)3, by the inductive hypothesis. Therefore, dim.(M/Mn) » cardinality [w(M) ~ (Mn) 3 . Thus, M/M © k, where K n | cardinality I =» cardinality [w(M ~ 0) ~ (M ~ 0 )], and therefore M/Mn and © (k,+) are isomorphic as additive groups. By A. 1 , 1 n n (l+M )/(l+M n) and M/M are isomorphic groups. Thus (I*M )/(1+M ) and © (k,+) are isomorphic groups. I Finite Rings. A.7. LetZ denote the ring of integers. Then (i) U(Z/(p°)) fa Z/(p -l) x Z /(p n ') , where p is a prime = 3^ (li) U(Z/(2n)) w (o}, if n = I; Z/(2), if n = 2; Z/(2n' 2) x Z(2), if n >2. (N. Bourbaki, Algebre, chapitre 7, Theoreme 3, page 78.) 70 A.8. For every integer k = 0, (l+p)^ ^ 1 + (mod p^+^ ). where p is a prime• « = > 3 i ; 5 =1+2 - i j. i ( mod . -)k+3\ Z ) . (N. Bourbaki, A1gebre. chapitre 7, Lemme 3, page 77.) A.9. Let R be a fin ite local ring with maximal ideal M. Then (i) cardinality R* = (card R/M) (card(R/M) ) where £(M) denotes the length of a composition series of M as an R-module. (ii) R* = (R/M)* x (l+M). ( i i i ) card M = (card R/M)^*^ , ([1*0, page 177) . A.10, Let k= GF(pn) , the finite field of pn elements. Then the only j* subrings of k are subfields GF(p ) , where r divides n. Proof. Since k is a fie ld ,any subring of k is a domain. A fin ite integral domain is a field. (Let R be a subring of k; consider the products ay, where a ( R is fixed and y varies over elements of R. ay = ay1 implies y = y1, since R is a domain. Thus, by the finiteness of R, ay = 1 for some y.) The result now follows from ([9 ], page 185, Theorem 13). An Isomorphism Theorem for Rings. A ,11. If L is a subring of a ring R, and N is an ideal of R, then the residue class ring L /L D N is isomorphic to the subring (L+N)/N of the residue ring R/N, ([1 6 ], Vol. 1, Theorem 8, page 14*0. Quadratic Extensions of the Rationals. A ,12. Let R1 denote the integral closure of R = in a quadratic extension K of the field of rational numbers. Then 71 (1) the factorization of R'p into prime ideals of R' is either (i) R'p = P with [R'/P : Z/(p)] = 2, or (ii) R'P = P2 with R'/P - Z/(p), or (iii) R'p = P]P2 with R'/P, = R'/P2 = Z/(p) . (2) is a discrete rank one valuation ring with group of divisibility G order isomorphic to the to ta lly ordered group of integers Z. The index of G in a group of divisibility of a valuation ring extending Z ^ j in K is one in case R'p = P or R'p = P,p 2 > ant^ ' s two 'n case R'p = P2 . (3) R1 is either a valuation ring or the intersection of two valuation rings. ([16], Vol I, page 287). (4) For each prime p, there exist quadratic extensions of the rationals such that (i), (ii), (iii) of (1) hold. (This follows from ([16], Vol. I, Theorem 32, page 3 1 3 ).). (5) The subrings R + R'pn of R' are all of the subrings of Rl containing Z(p) . (Unpublished notes of R. L. Pendleton). A,13. Let R' be the integral closure of in a quadratic extension of the rationals. Then U (R '/R 'pn) sa Z/(p2- l ) , if R'p = P, p = 2, n = 1; Z/(p2-l) x Z/(pn~’) x Z/(pn"'), if R'p = P, p = 3, n ^ 2 ; Z/(3) x Z/(2) x Z/(2) x Z /(2 n"2) x Z /(2n_2) , if R'p = P, p= 2, n = 2 ; Z/(p-l) x Z/(pn) x Z(pn), if R'p =P2,P = 3 ,n = 1; Z/(2) x Z/(2) x Z/(2n_1) x Z/(2n-1) , if R'p - P2, p = 2, and R'ff = P with tt2 t Z ^ j , n = 1 ; Z/(2n) x Z/(2n_l) , if R'p = P2 , p = 2 and R'tt = P with it2 e Z ^ , n = 1 ; Z/(p-l) x Z/(p-l) x Z/(pn"]) x Z/(pn“T), if R'p = P,P2, p = 3, n ^ 1 ; [0 ], if R'p = PjP2 > p = 2, n = 1; Z/(2) x Z/(2) x Z /(2n"2) x Z /(2 n_2) , if R'p = P1P2, p = 2, n = 2 . (P, P j, ?2 denote maximal ideals of R1 lying over (p),). Proof. If R'p = PjP2 1 then R' has exactly two maximal ideals Pj , P 2 . Then R'/R'pn = R'/(P]P2)n «R '/p" x R'/p" canonically and U(R' /R1 pn) » iKR'/?") x U(R'/P2) . Since R '/P ., i = 1,2, is fin ite , we can apply A,9 ( i i ) to each of the local rings R '/P1? . From A.9 and A. 12, we have the following: U(R'/R'pn) * Z/(p2-!) x (1+R'p)/(1+R'pn) , if R'p = P ; U(R'/R;pn) * Z/(p-l) x (1+P)/(1+R'pn) , if R’p = P2 and p = 3 ; U(R'/R'pn) «( 1+P) / ( 1+R' pn) , if R'p = P2 , p = 2; U(R'/R'pn) „ Z/(p-l) x Z/(p-1) x (l+P1P2)/(l + (P|P2)n), if R'p ** P1P2, U(R'/R'pn) « (l+P,P2)/(l+(P,P2)n) , if R'p = P1P2, p = 2 . Thus, we must determine the second factors of U(R'/R'pn) . From A . 9 ( i ) , we have the following: card U(R'/R'pn) = (p2- 1 ) .(p2) n * , R'p = P; card U(R1/R 1pn) = (p -1 )- (p )2n 1 , R'p = P2 ; card U(R'/R'pn) =(p - 1 )•(p )n-1 ■(p-1) • (p)n_1 , R'p= P,P2 . Thus, card (1+R»p)/ ( 1+R' pn) = p2n"2 , if R'p = P ; card (1+P)/(1+R'pn) = p2n_1 , if R'p » p2 ; card ( 1+R'p)/ ( 1+R'pn) = p2n if R'p = P,P2 . Suppose fir s t that p = 3 . I f R'p = P or R'p = P,Po, let v = ftp, where (1 f U(R’ ), 77 e R'p, _ n n n 77 * R'p . (1 + pp)p = 1 + pn. ^ + _ . + (P ) (^tp) r+ . . . + (yp)p . t Write r = p . s where p and s are re la tiv e ly prime. Then ( P is a 1 P m ultiple of pn *, and since pt > t. (fjp) r C R'p + . Thus, (r ) (fjip) r n+1 n . n-1 . Therefore, (] + JUp)P ■ 1 (mod p° ) () + Jip)P / I (mod p'1 ) , t « n-l y n-l because, for r > I, p = pP' S e R'p ; thus, (p2 ) (^p) + .,. + (ji p)p n+1 n~* n C R1pn , and hence (] + fip)P = 1 + p (^ + p a ), a e R' , implies (ii + pa f U(R1) since fx e U(R') and pa e j(R ‘ ), the Jacobson radical of R' . Thus, we conclude that every element of (1 + R’ p )/(l + R * pn) is of order pn” ^ or p' where i < n — 1 . 2 ------Suppose that R'p = p By A.8, 1+p generates a cyclic group of l n n order pn in ( 1+P)/ ( 1+R'pn) . Let it t P . (l+w)p = i + pnfl+.,.+ n n ( Pr ) irr+ ...+ ffP . Arguing as in the previous paragraph, we obtain n-l (1 +rr)P s 1 (mod pn) , (l+ff)P £ 1 (mod R'pn) . Thus, ( 1+P) / ( 1+R' pn) has a cyclic subgroup of order n-l and a cyclic subgroup of order n. Now assume that p = 2 . If n = 1, the result is immediate from A.9 and the fact that the group of units of a fin ite fie ld Fisa cyclic group of order pn — 1 , where pn is the cardinality of F, Assume n = 2 . 2 If R'p = P or R'p = Pjp2> 1 v = /ip, where fx € U(R') n i R'p . O m n nH“2 (I+t ) =| + 2n +...+ ( r )(ir)+...+(?r) n «n~2 n a a H** 1 «n-l = 1 + 2 V +...+ ( r ). n . 2 +.. .+(u . 2 ) e 1 + R'p" , p = 2 . 2 2n""^ n t (1 + n ) t I + R'p : write r = p .s, where 5 is odd, p = 2 , t n 3 For r > 1, pr = pP ,S , (I + ff2) 2 = 1 + 2n“T \x + 2n . a, as in the p = 3 case . 1 - 2 = -1 generates a cyclic subgroup of order two in 74 (I + R' p) / (1 + R'pn) . “T = (I + ff2)-* implies j = 2n ^ , and hence that 2 2 n - ( I + tr ) e 1 + R'p , which is a contradiction for n > 2. For n = 2 , 2 2 let 1 + ap e (l + R'p)/(1 + R'p ) . (1 +ap) = 1 + 2ap + a p ■ I Cp ) 2 for p = 2 . Hence, every element j4 1 of (1 + R‘p)/(1 + R'p ) is of ^n- 2 « ?n~2 «n—2 order two. (1+2) =1+2 . 2+.,,+{ r )2 +.,,+2 . 0n-2 ~n-2 «n-2 9 n -] n ( r ) 2r c R1pn for r > 2, p = 2, and ( 2 )2 e R'p , i R'p , p = 2 . Thus, (1 + 2 )2° 2 = 1 + 2n_1(l + (2n"2-l) + 2c) for some c, n > 2 , -n-2 „n-2 . implies (1 + 2) el+R'pn ; (1 + 2 ) e 1 + R'p . Moreover, f + 2 ^ 1 + ir^ . Thus, (l + R1 p)/ ( I + R'pn) has two subgroups of order 2n-2 2 2 Now suppose R'p = P , p = 2 . Suppose that IT e . By {[16], Vol. i, Theorem 32, page 313, and Theorem 34, page 317), we can take TT — 1 + /d where 2 does not divide d and d is square-free . Then ,n -1 , 1 + tt = /d i s of order 2 in (l+P)/(l+R‘p ) ; (1 + 2/d) is of order 2n_1 in { !+P/{l+R1pn) . 2n - 1 = “ = 1^2 and 2h" !-l = l+2(2n_2-l) each generate groups of order two in ( 1+P)/ ( 1+R1pn) . 2 2n 2n |* 2n Suppose tr e ^(p) * ^ en (1 + *0 = 1 + 2rV+ ...+ ( r)7r+...+ft = 1 + ff2N=1+ . . .+ (2r)ffr+ . . ,+ n 2 e P2n+iC. p2n = R'Pn , / R'Pn+1 , ! + 2tt i s of order 2n * in ( l+P/( 1+R1 pn) . 1 + 2ff = (1 + Tr)^ implies j = 2; then I + 2ff = 1 + 2 rr + n 2 implies 2 f R'pn , which is contradictory 75 2 2 2 f o r n > 1 . Ifn=l, then (1 + Jitr) = I + 2ppr + p tr = 2 ------1 + 2 u(ff + 2u) (ff = 2) implies 1 + uTT i s of order two, for *iny fi <■ R' , in {1 + P)/(l +R*p) , VITA Dolores Richard Spikes was born on August 24, 1936, in Baton Rouge, Louisiana, She attended the parochial and public schools of Baton Rouge. In 1957, she received the Bachelor of Science Degree from Southern University and in 1958 she received the Master of Science degree in mathematics from the University of Illinois. At present she is a candidate for the degree of Doctor of Philosophy in the department of mathematics at Louisiana State University. 76 EXAMINATION AND THESIS REPORT Candidate: Dolores Richard Spikes Major Field: Mathematics Title of Thesis: Semi-valuations and Groups of Divisibility Approved: M ajor Professor and Chairman . - - ______Dean of the Graduate School EXAMINING COMMITTEE: y t'v U v U U l. Date of Examination: 11 - 17-1971