DOCSLIB.ORG

Stable Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stable Rings Jcwnulof Acre md Applied Algebra 4 (I 974) 31 P-336. (Q North-Holland Publishing Company STABLE RINGS Judith D. SALLY Northwestern University, Evunston, Ilk 60201, U.S.A. and Wolmer V. VASCONCELKX * Rurgers Universiry, New Bmnswick, NJ. 08903, U.S.A. Communic%tcd by f-f. Bass Received 18 October 1973 Contents 0. Introduction 3iY 1. Estimates t’or numbers of generators of ideals 320 2. Stability 323 3. Twogmcrated rings 327 4. Non-Noetherian “two~enerated” rings 333 5. Non-standard stable rings 334 References 336 0. hwoduction One of the objectives of this paper is the examination of the relritionship between the following properties of a Noetherian ring A : (i) every ideal of2 can be generated by two elements; (ii) every ideal of A is projective over its endomorphism ring. Bass [3] in his study of the rings for which every torsion-free module is a direct sum of ideals, proved that if A is a t dimensional reduced ring with finite integral closure, then (i) implies (ii). Using properties of the canonical ideal, our i%st obser- vation in [22] - which we have since learned was proved earlier by Drozd and KiriEenko [9] by direct methods -.- was that, under the same conditions,(i) and (ii) are equivalent. Fn this paper we will show (Sections 3,5) that (i) and (ii) are not equivalent - even for domains - but that mild conditions on A will give (i) implies (ii). * Partially supported by National Science Foundation Grant 33 133. 319 Study of ctjndition ii) leads to more general questions involving estimates for bounds on the number of generators of certain ideals (Section I ‘) and to an exis- tence theorem far rings s;itisfying (i) (Section 3). We call a Noetherian ring stoblc if tt satisfies (ii). In Section 2 we show that stable rings have (Krull) dimension at most f and th3t every I dimensional local Xlar=aulayring possesses “a high degree sf stability”; to be spe&k, there exists an integer N such that, for all ideals f which ajntain a nunzero divisor. Psi is prczjective over its endomorphism ring. Sectkxn 4 is 3 digression on non-Nfoetherian rings. It contains the proof that finitely generated ideals in Priifer domains of dimension 1 are two-generated. Throughout e A is a <:ummutative ring with 1. The term “1~31” includes Noetk- rian. The f;Fliowing notatkjn will be fixed. k(B) denotes the length af an ,&module B, tql) denotes the minimal number of generators uf an ideal I and. if I has dimen- sivn 0. v(r) denotes the multiptkity of 1. &I ) denotes the multipkity of the Jacab- son radial of It. We wish to express our 3ppre+zi3ticlnto E. Matlis for his helpful comments. 1. Estimates fat numbers of generators of ideals One of the key tools in our study is the 1dimensional case if 3 theorem of Rees 12I i that gives a bound fc,r the number of generators of certsin ideals in a local Mac3ulay ring. First. we give a new. simple proof af Rees’ theorem and then several applications. Proof. Ler M be the maximal idea1 of A. By passing to A(X), W&V) and IA&n as In [I% pp. l&71], we may assume that A,W is an infinite field. The proof is by induction on (i, the dimension of A. Assume d = 1. ,4/M infinite implies that &f has 3 superficial element of degree 1 r Le., there is 3 nonzero divisor x iu M such that .tiP = /#It +1 for some t > 0. We have X(&k4 ) = h(r/xl) = p(A) . Far X(AjxA ) = X(AjxA) + X(xA/‘xi=) - X(A/i) = X(A/xl) - X(AII) = h(f,kf) gives the first equality, and, if we take / = IMk for any k 2 t, we have XIA/~A)=X(MklMk+‘)=c((A). The exact sequence 321 gives Assume d > 1. A/M infinite and A/J Mac3ulay imply that there exists a nonzero divisor x in M with the properties that x is a superficial element of degree t of A and x is not in any prime minimal over I; cf. [ 19, (22X)]. Then xl = xA n I, and is exact. Now A/xA is a Macaulay ring of dimension d - 1. The choice of x ensures that I/x1 is a height 1 ideal in A/xA and that A/(1, x) A is Macaulay. Thus, by induc- tion, u(i) = u(l/xl) < p(A/xA) = p(A) . q As a coroHary we obtain a bound on the number of generators for ideals in a semilocal ring#of dimension 1 (cf. (2,6,4,X, 121). We wJ1 write u(A) = n if every ideal of A can be generated by N elements and some ideal of A cannot be generated by N - 1 elements. Corollary 1.2. There is u bound an the number ofgenerators for kieals in a 1dimen- sional semilocul ring A. In particular. ifall maximal ideals of A are of height 1, then t--l v(A)%A/N) ki (1 +v(A@)<cr(A) n (I + tiNi)), j=l j=l where N is the nilradical of A, and t is the smallest integer such that N’ = (0). Proof. Rem11t.hat p(A) = Xp(A&, where M ranges over all maximal ideals of height 1. Since it is clear that a O-dimensional local ring has a bound for the number of generators of all ideals, it is suftkient, by the Forster-Swan theorem [23] , to prove the second statement of the corollary for A a l-dimensional local ring with maximal ideal N. The proof is by induction on t. if t = 1, i.e., if N = (0), A is Macaulay and I fi (0 :A I) = (0) for all ideals I of A. We apply Theorem 1 .l to the ideal I+(O:, r)=f@(O :A f’) and conclude that tr(l) <, p(A). ksume t > 1. From the exact sequence I O-+N’-’ _*,a# -+A/h+ +(J it follows that By induct ion, Example 1.3. Let x and )’ be indeterminates over a field k, and Iet A =k[fx.+~)]/(x~,~~$ ‘Ihen &I) = I , but the bound &4/N) (1 + v(!V)) = 2 obtained in Corallary f 2 is attained. Example 1.4. Here is a simple example of a f dimensional Noetherian ring with no bound on the number of generators for ideals (cf. [B, 7) .) Let -Ye,x2, .x3 . be infinitely many indeterminates over a field k. JLet The ideals Pi = (xi”, xi’*, ._., xfwl )R are height 1 prime ideals of R. Let S =R -* UPi and ,4 =Rs. A smali computation shows that if a prime ideal Q af R is contained in UP,, then Q is contained in some Pj. Hence (0) and the Pi ate the drily prime ideals sf A. Since ail the prime ideals of A are finitely generated, 4 is Noetherian, I-dimensional and manifestly without bound on the number icsfgener- ators for ideals. Our f&al application of the one-dimensional case of Theorem 1.1 is the follow- ing. bf* it is sufficient to show that there is a number n such that u(p) 5 n for all prime ideals P # M, the maximal ideal of A. IRt x 1, x2 be a system of parameters fur A, iwe.,M is minhnat aver (X1, ~2)~ If P is any prime ideal of A, P #M, then (Xf, x2) !$ P so, say x1 $ P. We have that 0 + P/x 1P + A/.x 1A is exact, hence u(P) = v(P/x*P)I H1 : where q is the bound obtained in Corollary 1.3 for the local ring A/.x 1A. If n2 is the bound fsr A/qA, then we rnpy take n = max(q. nr)* tl Note that the same prmf shows that P-primary ideals, P f M, have a bounded number of generators. J.D. Sdly, W. V. Vascmcelos, Stable rings 323 Remark 1.6. Macaulay’s examples (cf. [ 1] ) show that Corollary 1 S is no longer valid for higherdimensional local rings. It is afso not valid for global Noetherian rings of dimension 2. Remark 1.7. If A is a Zdimensional affine ring, there is a bound for the number of generators of prime ideals. Roof. We may assume that A is a domain finitely generated over the field k. By the Noether normalization theorem, there exist elements X,J’ in A, algebraically inde- pendent over k, such that A is a finite k [x. y] -module. Consider a presentation k[X.,?] n --z A + 0 of A9 and let P be a prime ideal of A. (We may assume that P is not maximal since, by the Nullstellensatz, a bound already exists for those.} Consider the exact sequence @-+(P)-+k(u - . y]” -4/P-+0 . Clearly, the projective dimension of A/P over k [x,y] is at most 1. Thus TV -I (P) is a projective and, therefore, free k I-u,y] -module of rank <, n. 2. St&My In analogy with LipmbJs terminology in [ 1S] , we call an ideal I of a ring A stable rf I is projective over its endomorphism ring End,(r). The ring A is stable if every ideal of A is stable. Note that if A is Noetherian and B is a faithfully flat A-algebra. then I is stable in A if and only if ,& is stable in B. Also note that if A is Noetherian, then I is stable in A if and only if /A, is stable in A,, for all maximal ideals M of A.
Load more

Recommended publications
  • Exercises and Solutions in Groups Rings and Fields
    EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets.
    [Show full text]
  • The Structure Theory of Complete Local Rings
    The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings.
    [Show full text]
  • Notes on Ring Theory
    Notes on Ring Theory by Avinash Sathaye, Professor of Mathematics February 1, 2007 Contents 1 1 Ring axioms and definitions. Definition: Ring We define a ring to be a non empty set R together with two binary operations f,g : R × R ⇒ R such that: 1. R is an abelian group under the operation f. 2. The operation g is associative, i.e. g(g(x, y),z)=g(x, g(y,z)) for all x, y, z ∈ R. 3. The operation g is distributive over f. This means: g(f(x, y),z)=f(g(x, z),g(y,z)) and g(x, f(y,z)) = f(g(x, y),g(x, z)) for all x, y, z ∈ R. Further we define the following natural concepts. 1. Definition: Commutative ring. If the operation g is also commu- tative, then we say that R is a commutative ring. 2. Definition: Ring with identity. If the operation g has a two sided identity then we call it the identity of the ring. If it exists, the ring is said to have an identity. 3. The zero ring. A trivial example of a ring consists of a single element x with both operations trivial. Such a ring leads to pathologies in many of the concepts discussed below and it is prudent to assume that our ring is not such a singleton ring. It is called the “zero ring”, since the unique element is denoted by 0 as per convention below. Warning: We shall always assume that our ring under discussion is not a zero ring.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • RING THEORY 1. Ring Theory a Ring Is a Set a with Two Binary Operations
    CHAPTER IV RING THEORY 1. Ring Theory A ring is a set A with two binary operations satisfying the rules given below. Usually one binary operation is denoted `+' and called \addition," and the other is denoted by juxtaposition and is called \multiplication." The rules required of these operations are: 1) A is an abelian group under the operation + (identity denoted 0 and inverse of x denoted x); 2) A is a monoid under the operation of multiplication (i.e., multiplication is associative and there− is a two-sided identity usually denoted 1); 3) the distributive laws (x + y)z = xy + xz x(y + z)=xy + xz hold for all x, y,andz A. Sometimes one does∈ not require that a ring have a multiplicative identity. The word ring may also be used for a system satisfying just conditions (1) and (3) (i.e., where the associative law for multiplication may fail and for which there is no multiplicative identity.) Lie rings are examples of non-associative rings without identities. Almost all interesting associative rings do have identities. If 1 = 0, then the ring consists of one element 0; otherwise 1 = 0. In many theorems, it is necessary to specify that rings under consideration are not trivial, i.e. that 1 6= 0, but often that hypothesis will not be stated explicitly. 6 If the multiplicative operation is commutative, we call the ring commutative. Commutative Algebra is the study of commutative rings and related structures. It is closely related to algebraic number theory and algebraic geometry. If A is a ring, an element x A is called a unit if it has a two-sided inverse y, i.e.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title One-sided prime ideals in noncommutative algebra Permalink https://escholarship.org/uc/item/7ts6k0b7 Author Reyes, Manuel Lionel Publication Date 2010 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Tsit Yuen Lam, Chair Professor George Bergman Professor Koushik Sen Spring 2010 One-sided prime ideals in noncommutative algebra Copyright 2010 by Manuel Lionel Reyes 1 Abstract One-sided prime ideals in noncommutative algebra by Manuel Lionel Reyes Doctor of Philosophy in Mathematics University of California, Berkeley Professor Tsit Yuen Lam, Chair The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplan- sky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bi- jectively to the classes of cyclic right R-modules that are closed under extensions.
    [Show full text]
  • Dedekind Rings Remark in Principal Ideal Rings All Non-Zero Prime Ideals Are Max- Imal Ideals
    Dedekind Rings Remark In principal ideal rings all non-zero prime ideals are max- imal ideals. We introduce a few properties on localizations which will be used later. Lemma 0.1. If the ring R is noetherian and p is a non-zero prime ideal of R then also the localization Rp is noetherian. a Proof. Let a˜ be an ideal of Rp. It is of the form R\p with an ideal a of R. The ideal a is finitely generated, say with generators ai ∈ R (1 ≤ i ≤ n). Clearly, the same elements generate a˜. 2 Lemma 0.2. Let R be a ring with prime ideals p, q subject to p ⊂ q. Then the corresponding ideals p q p˜ = , q˜ = R \ q R \ q are prime ideals of Rq satisfying p˜ ⊂ q˜. Proof. We start to show that prime ideals p of R which are contained in the prime ideal q satisfy p˜ ∩ R = p from which the second statement immediately follows. Clearly, we have p ⊆ p˜ ∩ R. On the other hand, any element of p˜ is of the form π/s with π ∈ p, s ∈ R \ q. If it is contained in R as well we obtain π = rs for some element r ∈ R. Since s is not contained in p we must have r = π/s ∈ p, hence, p˜ ∩ R ⊆ p. Next we show that p˜ is a prime ideal of Rq. If the product of two elements r/s and u/v of Rq is contained in p˜ we get ru ∈ p, hence either r ∈ p (with the consequence r/s ∈ p˜) or u ∈ p (with the consequence u/v ∈ p˜).
    [Show full text]
  • CHINESE RINGS Karl Egil AUBERT and Istvan BECK in a Commutative
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 24 (1982) 221-226 221 North-Holland Publishing Company CHINESE RINGS Karl Egil AUBERT and Istvan BECK Department of Mathematics, University of Oslo, Blindern. Norway Communicated by H. Bass Received 4 April 1981 1. Introduction In a commutative ring R with an identity element one can consider a multiplica- tive congruence which is coarser than the classical congruence modulo an ideal a in R. Forming the factor ring I? = R/a we declare b and c in R as canonically congruent module a whenever these two elements give rise to residue classes 6, c which generate the same principal ideal (6) =(c) in R. We use the term ‘canonical’ because this con- gruence may be characterized as the unique coarsest multiplicative congruence on R with the property that any ideal b > a is a union of congruence classes. We shall denote the canonical congruence modulo a by b = c (a) whereas the classical con- gruence is, as usual, denoted by b =c (mod a). Specializing to the ring E of integers we have the following suggestive interpreta- tion: the integers a and b are canonically congruent modulo n iff the greatest common divisor of a and n equals the greatest common divisor of b and n. The aim of the present paper is to investigate for which rings the Chinese Remainder Theorem holds for a finite collection of canonical congruences which are compatible in an obvious sense.
    [Show full text]
  • Chapter II. Rings and Polynomials
    Matematisk Institut Mat 3AL 2.1 Chapter II. Rings and polynomials PREREQUISITES ABOUT RINGS AND IDEALS. A ring R is a non-empty set with two compositions + and · such that 1) (R, +) is an abelian group; 2) (R, ·) is associative; 3) The distributive laws hold: a · (b + c)=a · b + a · c (a + b) · c = a · c + b · c. The neutral element in R with respect to addition + is denoted by 0. A neutral element with respect to multiplication · is uniquely determined and is called the identity element or unit element and is usually denoted by 1 (or sometimes by e). If (R, ·) is commutative, R is called commutative In the following we shall only consider commutative rings with identity element. We recall some basic definitions and theorems which should be known from Anders Thorup Mat 2ALG. For any element a of R the distributive law implies: a · 0=a · (0 + 0) = a · 0+a · 0, hence a · 0=0.Thusa · b = 0 if either a =0orb =0. A commutative ring for which the converse holds, i.e. a commutative ring for which the product of two elements a and b can only be 0 if either a =0orb =0,is called an integral domain. A commutative ring R (with at least two elements) is called a field, if every non- zero element a in R has an inverse, i.e. an element a−1 ∈ R such that a · a−1 =1. Such an inverse is necessarily uniquely determined. (Why?) A field is necessarily an integral domain. Indeed, assume a =0and a · b =0.Ifa has an inverse a−1 the equation a · b = 0 implies 1 · b = a · a−1 · b = 0; hence b =0.
    [Show full text]
  • Principal Ideals in Matrix Rings
    JOURNAL OF RESEARCH of the Notional Bureau of Standards - B. Mathematical Sciences Vol. 73B, No.3, July-September 1969 Principal Ideals in Matrix Rings Morris Newman and Stephen Pierce Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (April 21, 1969) It is shown that every left ideal of the complete matrix ring of a given order over a principal ideal ring is principal, and a partial converse is proven. Key words: Dede kind ring; matrix ring; non-Noetherian ring; principal ideal ring. 1. Introduction Let R be a ring with a unity 1, and let n be a positive integer. It is well-known [3, p. 37]1 that every two-sided ideal of R" (the complete matrix ring of order n over R) is necessarily of the form M", where M is a two-sided ideal of R. Simple examples show that this result no longer holds for one-sided ideals. In this note we investigate the left ideals of R" in the case when R is a principal ideal ring (an integral domain in which every ideal is principal). We shall prove THEOREM 1: IfR is a principal ideal ring, then every left ,ideal ofRn is principal_ The proof of Theorem 1 depends upon the fact that if A is any p X q matrix over R, then a unit matrix V of Rp exists such that the p X q matrix VA is upper triangular [2 , p. 32]_ We also establish the following partial converse to Theorem 1: THEOREM 2: If R is not Noetherian or if R is a Dedekind ring but not a principal ideal ring, then Rn contains a nonprincipalleft ideal_ For general information on rings, see [3].
    [Show full text]
  • Arxiv:1812.04326V2 [Math.KT] 25 Jun 2019 Xeddt H Aeo Stoi Ak1
    CHEVALLEY GROUPS OF POLYNOMIAL RINGS OVER DEDEKIND DOMAINS A. STAVROVA Abstract. Let R be a Dedekind domain, and let G be a split reductive group, i.e. a Chevalley– Demazure group scheme, of rank ≥ 2. We prove that G(R[x1,...,xn]) = G(R)E(R[x1,...,xn]) for any n ≥ 1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G = SLN , Sp2N . We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R. 1. Introduction A. Suslin [Su, Corollary 6.5] established that for any regular ring R of dimension ≤ 1, any N ≥ 3, and any n ≥ 1, one has SLN (R[x1,...,xn]) = SLN (R)EN (R[x1,...,xn]), where EN (R[x1,...,xn]) is the elementary subgroup, i.e. the subgroup generated by elementary matrices I + teij, 1 ≤ i =6 j ≤ N, t ∈ R[x1,...,xn]. In particular, this implies SLN (Z[x1,...,xn]) = EN (Z[x1,...,xn]). A later theorem of A. Suslin and V. Kopeiko [SuK, Theorem 7.8] together with the homo- topy invariance of orthogonal K-theory (see [Kar73, Corollaire 0.8], [Hor05, Corollary 1.12], or [Sch17, Theorem 9.8]) implies a similar result for even orthogonal groups SO2N , N ≥ 3, un- der the additional assumption 2 ∈ R×. F. Grunewald, J. Mennicke, and L. Vaserstein [GMV91] extended the result of Suslin to symplectic groups Sp2N , N ≥ 2, and a slightly larger class of rings R, namely, locally principal ideal rings. One says that a (commutative associative) ring A with 1 is a locally principal ideal ring, if for every maximal ideal m of A the localization Am arXiv:1812.04326v2 [math.KT] 25 Jun 2019 is a principal ideal ring.
    [Show full text]
  • On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics
    On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics∗†‡ M. Behboodia,b§and A. Moradzadeh-Dehkordia aDepartment of Mathematical Sciences, Isfahan University of Technology P.O.Box: 84156-83111, Isfahan, Iran bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM) P.O.Box: 19395-5746, Tehran, Iran [email protected] [email protected] Abstract In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring (R, M), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M = Lλ∈Λ Rwλ and R/Ann(wλ) is a principal ideal ring for each λ ∈ Λ; (3) Every prime ideal of R is a direct sum of at most |Λ| cyclic R-modules; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring (R, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M. arXiv:1202.0371v1 [math.AC] 2 Feb 2012 1. Introduction It was shown by K¨othe [8] that an Artinian commutative ring R has the property that every module is a direct sum of cyclic modules if and only if R is a principal ideal ring.
    [Show full text]