Describing Ideals of Endomorphism Rings 1

Home , Endomorphism ring

DESCRIBING IDEALS OF

ENDOMORPHISM RINGS

Brendan Goldsmith and Simone Pabst

Introduction

It i s well known that the r ing of linear transformations of a nite

dimensional vector space i s s imple ie it has no nontr ivial proper

twoside d ideals It i s p erhaps not so well known that the two

s ide d ideals in the r ing of linear transformations of an innite

dimensional vector space can b e characterize d by a s ingle cardinal

invariant It i s therefore re asonably natural to ask if there i s a

generalization of Baers re sult to ideals in the endomorphism r ing

of a wider clas s of mo dules The purp o s e of this pre s ent work i s

to explore this p o s s ibility

A rst generalization i s to replace the underlying eld of the

vector space by a r ing A natural extension of the concept of a

eld i s a di screte valuation r ing s ince a di screte valuation r ing

mo dulo its Jacob son radical i s a eld Recall s ee eg that

R i s a di screte valuation r ing if R i s a pr incipal ideal r ing with

exactly one maximal ideal or alternatively with one pr ime element

p In particular the r ing of padic rationals Z i s a di screte

valuation r ing and the re ader may replace all reference s to di screte

valuation r ings with the padic rationals without any s er ious lo s s

in generality Also recall that if E i s a r ing then the Jacob son

radical of E i s dened to b e the intersection of all maximal ideals

of E It i s well known that this i s equivalent to the s et of all

elements x E such that r xs i s quas iregular for all r and s in

E ie thos e elements x for which r xs i s a unit

Since a vector space over a eld F i s a f ree F mo dule an

obvious que stion to ask i s whether a corre sp onding characteriza

tion of ideals in the endomorphism r ing of a f ree mo dule over a

Ideals of Endomorphism r ings

di screte valuation r ing exi sts We addre s s this problem in x and

obtain Baers Theorem for vector space s as a corollary to our main

re sult Theorem

Whilst f ree mo dules over a di screte valuation r ing are an obvi

ous generalization of vector space s it i s p erhaps not so obvious

that complete mo dules over a complete di screte valuation r ing R

have many s imilar properties to vector space s cf Recall

that a complete di screte valuation r ing i s a di screte valuation r ing

which i s complete in its padic topology ie the linear topology

with bas i s of neighbourhoo ds of given by p R n where p

i s the only pr ime element of R In x we di scus s the ideal structure

of the endomorphism r ing of such mo dules and achieve a complete

characterization mo dulo the Jacob son radical

We conclude this introduction by reviewing a number of

standard concepts in mo duleabelian group theory If R i s di s

crete valuation r ing with pr ime element p then we say that an R

n n

submodule H of the R mo dule G i s pure in G if H p G p H

for all n If we cons ider the mo dule G as a topological mo dule

fur ni shed with the padic topology ie a bas i s of neighbourhoo ds

of i s given by p G n then H i s pure in G preci s ely if

the padic topology on H coincide s with the induced subspace

topology Also an R mo dule D i s divisible if given any d D

we can solve the equation p x d in D a mo dule i s reduced if

it has no nontr ivial divi s ible submodules It i s well known that

a torsionf ree divi s ible R mo dule i s a direct sum of copies of the

quotient eld Q of R Notice that if H G then GH divi s ible

i s equivalent to H b e ing dens e in the padic topology on G

Finally note that maps are wr itten on the r ight and the word

ideal will always mean a twoside d ideal

Preliminarie s

In this s ection we state a few wellknown re sults on mo dules over

complete di screte valuation r ings Throughout the s ection R shall

b e a complete di screte valuation r ing with pr ime element p In

cons idering torsionf ree R mo dules the concept of a bas ic sub

mo dule i s us eful a submodule B of a torsionf ree R mo dule A i s

calle d a basic submodule of A if B i s a f ree R mo dule such that

IMS Bulletin

AB i s divi s ible and B i s pure in A

The rst lemma i s due to R B Wareld it tells us how

to obtain a bas ic submodule f rom the R pR vector space ApA

Lemma Let A b e a torsionf ree R mo dule and A

ApA the natural epimorphi sm Let fx j i I g b e an R pR

bas i s of ApA and choo s e y A i I such that y x Then

i i i

the submodule B generated by fy j i I g i s a bas ic submodule of

A Moreover every bas ic submodule of A ar i s e s in this way

The lemma ensure s the exi stence of a bas ic submodule B and

the uniquenes s of its rank rkB where rkB i s the usual rank of a

f ree R mo dule Hence we may dene the rank of a torsionf ree R

mo dule A as the rank of its bas ic submodule ie rkA rkB

Since the divi s ibility of the quotient mo dule AB i s equivalent to

the density of B in A in the padic topology we obtain the following

wellknown characterization of complete re duced torsionf ree R

mo dules i s e s s entially due to I Kaplansky and i s

s imilar to a re sult proved for pgroup s by H Leptin

Propo s ition The following properties of a re duced torsion

f ree R mo dule A are equivalent

A i s complete

If B i s a bas ic submodule of A then A i s the completion of B

If B i s a bas ic submodule of A then every R homomorphism

B A extends uniquely to an R endomorphism of A

Next we state a few f acts on complete re duced torsionf ree

R mo dules the pro ofs can b e found in and

Lemma Let A and A b e complete re duced torsionf ree R

mo dules

If i s an endomorphism of A then b oth ker and im are

complete

If S i s a pure submodule of A and i s complete then S i s a

direct summand of A

0 0

A i s i somorphic to A if and only if rkA rkA

We ni sh this s ection with a standard re sult on the Jacob son

radical of the endomorphism r ing of a complete re duced torsion

f ree R mo dule a pro of may b e found in

Ideals of Endomorphism r ings

Propo s ition Let E denote the endomorphism r ing EndA of

the complete re duced torsionf ree R mo dule A Then

J E pE E p

End ApA E J E

RpR

J E f E j A pAg

Free mo dules over di screte valuation r ings

Here we shall di scus s ideals of endomorphism r ings of f ree R

mo dules over di screte valuation r ings R We will de duce Baers

Theorem on vector space s as a corollary to our main re sult Before

we re str ict our attention to mo dules over di screte valuation r ings

we prove a re sult Propo s ition which i s true for mo dules in

general The denition of a direct endomorphism will b e us eful

An endomorphism of A i s calle d kdirect imdirect if ker

Im i s a direct summand of A and i s said to b e a direct

endomorphism if it i s b oth kdirect and imdirect First we need

Lemma If i s a direct endomorphism of A then there exi sts

an endomorphism of A such that

a and are b oth idemp otent and

b Im Im Im Im ker ker and

ker ker

Pro of By our as sumption we may wr ite

A Im S and A ker T

Then the re str iction T Im i s an i somorphi sm Hence

there exi sts Im T such that

id and id

T T T

Now let A A b e dened by

and ker S

We shall s ee that i s the require d endomorphism

First we show that and are idemp otent Let x b e an

arbitrary element of A Then

x x x x

IMS Bulletin

s ince i s the identity on Im So Now let x

a s A where a A and s S Then

x x a a x

Thus as require d

To prove part b we make the following calculations

A A A A Im Im

A A T Im Im

ker fx Ajx g fx Ajx ker g

fx Ajx ker T g ker

ker fx Ajx g fx Ajx g

fx Ajx g ker

Thi s completes the pro of

Propo s ition Let I b e an ideal of the endomorphism r ing

EndA of A such that all I are kdirect Moreover as sume

that I contains a direct endomorphism If i s an imdirect

endomorphism of A such that Im i s i somorphic to a direct

summand of Im then b elongs to the ideal I

Pro of Let I and b e as above Then we can wr ite

A Im S Im T and Im R C

where R Im We extend to an endomorphism of A by

and

R C S

Now cons ider the endomorphism I Since b elongs to I it

i s kdirect Moreover

A R C R R Im

which i s a direct summand of A Hence i s direct and we may

apply Lemma to Thus there exi sts EndA such that

Ideals of Endomorphism r ings

i s an idemp otent and Im Im Im Therefore

for any x A there i s y A with x y Hence

x y y x

which implie s that i s in I s ince I i s an ideal

Now let R b e a di screte valuation r ing and A a f ree R mo dule

The next theorem tells us something about ideals I containing a

direct endomorphism of a given rank Recall that the rank of an

endomorphism i s dened to b e the rank of the f ree R mo dule

Theorem Let I b e an ideal of EndA If I contains a direct

endomorphism of rank then I contains all endomorphisms of

rank le s s than or equal to

Pro of First we show that all endomorphisms of A are kdirect

Let b e any endomorphism of A Then A ker Im

where Im i s f ree and hence pro jective Thus there exi sts a

homomorphism Im A with id We show that

A ker Im Let x ker Im Then there i s

y A such that x y and

x y y

s ince id and thus x y Also

a a a a with a a a a

for any a A again s ince id Thus ker i s a direct

summand of A implying that i s kdirect

Now let I b e a direct endomorphism of A of rank ie

A Im S and rkIm Next we prove that all direct

imdirect endomorphisms with rk are elements of I

Let b e such an endomorphism Then

A Im T and rk rkIm

Since Im and Im are f ree R mo dules with rkIm

rkIm there exi sts a direct summand of Im which i s i so

morphic to Im Thus we may apply Propo s ition which

implie s that I

IMS Bulletin

Finally let b e any endomorphism of A with rk

Since i s kdirect we may expre s s A as A ker C where

Im Thus A ker C

rkC rk

Let b e the pro jection of A onto C with ker ker Obvi

ously i s a direct endomorphism with rk rkC Thus

I Hence i s an element of I and this completes

pro of

Note that the previous theorem holds for f ree mo dules over

any r ing R having the property that submodules of f ree mo dules

are f ree eg all pr incipal ideal domains have this property So in

particular Theorem holds for a eld In this cas e we get even

more namely we can characterize the ideals of the endomorphism

r ing EndA of a vector space A

Corollary Let A b e a vector space over a eld R Then

the only ideals of EndA are the ideals E dened by

E f EndAj rk g

Pro of Note rst that all endomorphisms of a vector space A are

direct Hence in this cas e Theorem re ads as

If is an element of an ideal I then I contains every endomorph

ism with rk rk

It i s e asy to check that for e ach E i s an ideal of EndA

the ideal of all nite rank endomorphisms Write E for E

Now let I b e an arbitrary ideal which i s properly con

tained in EndA Since I i s nontr ivial there exi sts a nonzero

endomorphism I If i s of innite rank then obviously

E I So suppo s e i s of nite rank n In this cas e I con

tains all endomorphisms of rank le s s than or equal to n Thus if

e i s an element of a given bas i s B then I contains the pro jection

onto the onedimensional subspace generated by e along the

subspace generated by the remaining bas i s elements Therefore

e B of such pro jections b elong to I all nite sums

i e

and hence for any k N there i s an endomorphism of rank k

b elonging to I Thi s implie s that all endomorphisms of nite rank

are contained in the ideal I and so in e ither cas e we de duce that

Ideals of Endomorphism r ings

E I Moreover E I whenever I for some of rank

Let b e the smallest cardinal with E I we may cons ider

a succes sor cardinal s ince the ideals E form a smo oth incre as ing

chain Then all I have rank le s s than and hence I E

Also E I by the minimality of thus I E

Complete mo dules over complete di screte valuation

r ings

In the last s ection we turn our attention to complete re duced

torsionf ree R mo dules A over complete di screte valuation r ings

R Recall that the rank of a re duced torsionf ree R mo dule over

a complete di screte valuation r ing i s the rank of a bas ic submodule

B of A s ee Section Again we dene the rank of an endomorph

i sm as the rank of its image Moreover we call an endomorphism

of A a pure endomorphism if Im i s a pure submodule of A

First we pre s ent a re sult which i s s imilar to Theorem

for a complete re duced torsionf ree R mo dule A over a complete

di screte valuation r ing R we can prove

Theorem Let I b e an ideal of EndA If I contains a pure

endomorphism of rank then I contains all endomorphisms

with rk

Pro of Firstly we show that any endomorphism of A i s kdirect

By Lemma it suce s to show that ker i s pure in A for any

EndA If x p a with x ker and a A we have

n n

p a x Hence p a which implie s a s ince

A i s torsionf ree So

n n

p A ker p ker

that i s ker i s pure in A and thus i s kdirect for any

EndA

Let b e a pure endomorphism in I and as sume rst that

i s a pure endomorphism of A Then by Lemma b oth Im

and Im are direct summands of A ie we may wr ite A as

A Im S Im T

IMS Bulletin

If B B are bas ic submodules of Im and Im re sp ectively

then there exi sts a direct summand D of B of rank

rk rkB rkB rk

which i s i somorphic to B We may extend this i somorphi sm to

an i somorphi sm of the completions Im and D of B and D

re sp ectively Moreover s ince D i s pure in B the completion D

i s pure in Im hence D i s a direct summand of Im which i s

i somorphic to Im Thus we may apply Propo s ition which

implie s that I We have shown that all pure endomorphisms

with rk rk are contained in I So in particular all

idemp otents with rk rk b elong to I

Finally let b e any endomorphism of A with rk rk

Then A ker C where C A ker Im and so

rkC rk If denotes the pro jection onto C with ker

ker then I s ince

rk rkC rk rk

Therefore i s an element of I

The previous theorem however do e s not characterize the

ideals of EndA s ince there are ideals which do not contain a

pure endomorphism for example p EndA Instead of us ing s im

ilar arguments as in the cas e of f ree R mo dules we shall now us e

Corollary on vector space s to determine the ideals I of EndA

mo dulo their Jacob son radicals First we cons ider the Jacob son

radicals J E of the ideals E where

E f EndAj rk g

for

Lemma Let E b e an ideal as dened above Then the Jac

ob son radical J E coincide s with the ideal pE

Pro of First we show that pE E pEndA Let p E with

EndA Then A A kerp and rkA Since A i s

torsionf ree kerp ker Hence A A and rk

Ideals of Endomorphism r ings

Thus E By Propo s ition p EndA J EndA and so

it follows that

E p EndA E J EndA

But E J EndA J E s ince E i s an ideal of EndA

Therefore

pE E p EndA E J EndA J E

Thi s completes the pro of

Next we will show that E J E i s i somorphic to a corre s

p onding ideal of the vector space ApA over the eld R pR

Lemma For any cardinal E J E E ApA

Pro of Every E induces an R pR endomorphism on ApA

s ince pA pA So we may dene a map

E End ApA

RpR

ApA ApA with a pA a pA

It i s e asy to check that i s a r ing homomorphism Moreover the

kernel of i s pE J E We show that Im E ApA

Certainly Im E ApA s ince the vector space rank of an

endomorphism of ApA cannot b e gre ater than rk Now let

A ApA and R R pR b e the endomorphisms

dened by

a a pA and r r pR

Let us cons ider an endomorphism of ApA of rank le s s than

We can pick an R pR bas i s fx j i I g of ApA such that x

i i

for all but le s s than of the x Choo s e y A such that y x

i i i i

for all i I Then the mo dule generated by fy j i I g i s a bas ic

submodule of A by Lemma However

x r x

i ij j

i2I

IMS Bulletin

where r R pR with r for all but nitely many j Choo s e

ij ij

s R with s r and s whenever r Finally

ij ij ij ij ij

dene B B by

y s y

i ij j

i2I

so that y for all but le s s than of the y Thus the unique

i i

extension of to an endomorphism of A has rank rkB

and satise s Hence E E ApA i s surjective

with ker pE J E Thus E J E E ApA as

require d

We ni sh the pap er with the characterization of the ideals

of the endomorphism r ing of a complete re duced torsionf ree R

mo dule A over a complete di screte valuation r ing R mo dulo their

Jacob son radical they are characterize d by a s ingle cardinal

Theorem If I i s an arbitrary ideal of the endomorphism r ing

of a complete re duced torsionf ree Rmo dule A then e ither I

J EndA or

I J I E J E E ApA

for some cardinal

Pro of Let I b e any ideal of EndA We cons ider the mapping

I EndApA dened by

with a pA a pA

Thi s denes a r ing homomorphism with

ker J I I J EndA

which i s e ither equal to I ie I J EndA or i s properly

contained in I In the latter cas e I J I K for some non

zero ideal K of EndApA Thus K E ApA for some by

Corollary Therefore I J I E J E by Lemma

Ideals of Endomorphism r ings

Reference s

R Baer Linear Algebra and Pro jective Geometry Academic Pre s s

New York

L Fuchs Innite Abelian Group s vol I Academic Pre s s New York

L Fuchs Innite Abelian Group s vol I I Academic Pre s s New York

B Goldsmith S Pabst and A Scott Unit sum numbers of r ings and

mo dules to appe ar

I Kaplansky Innite Abelian Group s The University of Michigan

Pre s s

H Leptin Zur Theorie der uberabzahlbaren abelschen pGruppen

Abh Math Sem Univ Hamburg

W Liebert Characterization of the endomorphism r ings of divi s ible tor

s ion mo dules and re duced complete torsionf ree mo dules over complete

di screte valuation r ings Pacic J Math

R B Wareld Homomorphi sms and duality for torsionf ree group s

Math Z

Brendan Goldsmith and Simone Pabst

Department of Mathematics Statistics and Computer Science

Dublin Institute of Technology

Kevin Street

Dublin