Direct sums of dual continuous modules

Math. Z. 178, 225-232 (1981) Mathematische Zeitschrift

�9 Springer-Verlag 1981

Direct Sums of Dual Continuous Modules*

Saad Mohamed 1 and Bruno J. Miiller 2

Department of Mathematics, Kuwait University, Kuwait z Department of Mathematics, McMaster University, Hamilton, Ontario, Canada

To Professor Goro Azumaya on his 60th Birthday

1. Introduction

All rings considered have unities and all modules are unital right modules. For an R-module M, Rad M will stand for the Jacobson radical of M. If N is a small submodule of M, we write N c M. A module M is called dual continuous

s

(d-continuous) if it satisfies the following two conditions: (I) for every submodule A of M there exists a decomposition M = M I ( ~ M 2 such that M i c A and M 2 ~ A ~ M , and (II) every epimorphism from M onto a summand of M

splits. It follows from condition (I) that if M is indecomposable and d-continuous, then every proper submodule of M is small in M, and therefore either M = R a d M or M is local. It was proved in (1-6], Corollary 3.11) that an indecomposable d-continuous module has local endomorphism ring. These proper- ties of an indecomposable d-continuous module will be used often in this paper without further reference. A decomposition theorem for d-continuous modules was obtained by Mohamed and Singh [6], and was later improved by Mo- hamed and Mtiller [5] as follows:

Theorem (A). A d-continuous module has a decomposition, unique up to isomor-

phism, M = ~ @ A i O N where each A i is a local module and N = Rad N. i~I

It follows from the above theorem that a d-continuous module with small radical is a direct sum of local modules.

A summand of a d-continuous module is d-continuous ([-6], Proposition 3.5). However if M 1 and M z are d-continuous modules, then M I O M 2 need not be d-continuous. The following is proved in ([6], Proposition 4.1):

Theorem (B). I f M 1 0 M 2 is d-continuous, then M 1 is M2-projective.

We have been concerned with the converse question, namely whether the direct sum of d-continuous modules, which are projective with respect to each

* This research is supported in part by the NSERC of Canada, Grant A 4033

0025-5874/81/0178/02225/$01.60

226 S. Mohamed and B.J. Mtiller

other, is again d-continuous. In this direction, the following two results were proved in [5-1 and [8], respectively.

Theorem (C). I f M = ~ @ A i, where each A i is local d-continuous and A;-pro- t = l

jective for all j ~ i, then M is d-continuous.

Theorem (D). Let M = ~ @ A t, where each A t is local d-continuous and A/pro- iEI

jective for all j ~ i. I f Rad M ~s M, and if every pure sumodule of M is a summand, then M is d-continuous.

In the present work, we show that 'A t is local' in Theorem (C) can be re- placed by 'A t is indecomposable ' ; A t may then be equal to its own radical. We also prove that the conditions 'A t is local' and 'every pure submodule is a summand ' in Theorem (D), are superfluous. We then apply these results to local and to commutative rings.

The Krull-Schmidt-Azumaya Theorem, and the following results on relative projectivity (cf. [7, 2, 3]) are fundamental to our investigation:

Theorem (E). A direct sum of modules is M-projective if and only if each summand is M-projective.

(i ) Theorem (F). I f a module N is Mr-projective, 1 <-_i<n, then N is @ M t - projective. \t= 1

Theorem (G). I f a finitely generated module N is M~-projective, 2~A, then N is ( ~ @ M ~)-projective. 2~A

2. Main Theorems

We start by recording the following well-known lemma.

Lemma 2.1. Let A,B, C be R-modules. I f f: A ~ B and g: B ~ C are R-homo- morphisms such that g f is an isomorphism, then B = I m f | Kerg.

Now we prove our first main theorem. FI

Theorem 2.2. Let M = ~ @ A t. I f A i is indecomposable d-continuous and Aj- t = l

projective for all j ~ i, then M is d-continuous.

Proof Condition (I): Since each A i has local endomorphism ring, we deduce from the Krull-Schmidt-Azumaya Theorem that any summand X of M is of

t

the form X = ~ @ Xk, t_-< n, where each X k is isomorphic to some A t. Let A be k=l

a submodule of M. Consider the set of all summands of M contained in A and select one of maximal (direct decomposition) length, say M 1. Let M - - M 1 0 M 2 and let B = A c~M=. We prove that B c M . On the contrary assume that B is

s

not small in M. Let nt denote the natural projection of M onto A t. Then

Direct Sums of Dual Continuous Modules 227

B c @rhB. Since every proper submodule of A~ is small in A~, and B is not i = i

small in M, we get ~kB=Ak for some k.. Then

M = , + E| f~k

By Theorem (F), A~ is ( ~ @ A~)projective, and consequently there exists a ho-

momorphism r making the following diagram commute:

A k

/ / / / / ' nat

2 0 H i flat ~1. M/B

Let A'k={a--4)(a): aeAk}. It is clear that A'kcB and M=A'k| ~@Av But then

M~| is a summand of M contained in A having greater iength then M~, which is a contradiction. Tfierefore, condition (I) holds.

Condition (II): The proof is simiiar to that given in ([5], Theorem 2). How- ever we include it for the reader's convenience. First we prove that any epinaor- phism C - ~ D , from a summand C of M onto an indecomposable summand D of M, splits. Now D is isomorphic to some A , and C=~@C~ where

jeY J~{1,2 , ...,n} and Cj is isomorphic to Aj. Since every proper submodule of D is small in D, ~7(Ck)=D holds for some keJ.

Since A~ is d-continuous, and Ak-projective if k + r, the sequence

A k----'---~ C~ nat,..~ C ~ ) D -- ) A r

splits. Since A~ is indecomposable, the sequence is an isomorphism from A k onto .A~. But then

nat ('k --- ---~ C ~ D

is an isomorphism, and it follows by Lemma2.t that C=C~| holds. Hence the epimorphism C--2--~ D splits.

Next let B be a summand of M of length t (t<n-~ and let f : M-*B be an epimoi~hism. To prove that f splits, we use induction on t. Assume that any epimorphism from M onto a summand of length less than t splits. Write B=B~ @B 2 where B~ is a summand of length t - i and B 2 is an indecomposabte summand. Let e~ denote the natural projection of B onto B~, i=1,2, By as- sumption the sequence

M ~ B 1 ,0

splits. Then M--MI| where M2=Kerelf and elflM 1 is an isomorphism. Let g~ denote the natural injections of M~ into M, and let f~s=e~fgs; i,j= 1,2. Then f can be represented by the matrix

(,fi~ 0 f~ f:)"

228 s. Mohamed and B.J. Mailer

Define 0: B-~B by the matrix

It is clear that 0 is an isomorphism, and Of: M--,B is an epimorphism, Since

0), it follows that f~z: M2-~B2 is an epimorphism. By the result already proved, f22 splits. Let f~2 be the splitting homomorphism. Define qS: B-+M by the matrix

(0 Then

0 1

Hence Ms-:~,B ~ 0 splits. Thus condition (It) holds.

To prove our next main theorem we need the following:

Lemma 2.3. Let ApA2, . . . ,A n be local indecomposable summands of a d-con-

tinuous module M. I f A~8#Aj for all i=t=j, then ~ A~ is direct and is a summand of M. i=x

Proof, by induction over n. Assume that A I + A 2 + . . . + A , _ 1 is a direct sum and is a summand of M. Let N=A~|174 ~. By the Krull-Schmidt- Azumaya theorem, every indecomposable summand of N is isomorphic to one of the Ai, l<_i<_n-1. It then follows by ([6], Proposition 4.5) that N + A , is a summand of M. Now Nc~A,~A , implies that N n A , ~ M . However Nc~A,~ is

a summand of M by ([6], Lemma 3.6). Therefore N c~ A n = 0, and hence

A~ + A 2 + ... + A , = N + A~=N| ~.

Theorem 2.4. Let M be a module with small radical. Then M is d-continuous if and only if M = ~@A~ where Ai is local d-continuous and Aiprojective for all

Proof. The only if part follows by Theorem (A), Conversely let M be as in the statement of the theorem,

Condition (I). Let A be a submodule of M. Since M/Rad M - ~ @ AJRad A~, M/Rad M is semisimple. Hence

M/Rad M = (A + Rad M)/Rad M @ ~ @ AJRad Aj j e J

where J ~ I. Let M" = ~@') A~. Then M = A + Rad M + M". Since Rad M ~ M, we get J~J s

M = A + M " .


As Ai is cyclic, A~ is M"-projective for all i6J by Theorem (G). Therefore ~,@A~ is M"-projective by Theorem(E). Consequently, there exists a homo- id~J

morphism ~b: ~ @ A i - . M" such that the following diagram commutes iSJ

E| /~/ i6J i nat

M" , M/A , O. nat

Define M'={m-(a(m): m ~ @ A i } . Then it is obvious that M ' c A and M = M ' jC~J

| Since A c ~ M " c R a d M c M , and M" is a summand of M, we get s

A c~ M " c M". This proves that condition (I) holds. s

Condition (II). Let B be a summand of M and f : M ~ B be an epimorphism. Write M=B| Since each A i is cyclic and has local endomorphism ring, then by Warfield (I-7], Theorem 1), the decomposition M = B O B ' refines to one isomorphic to M = ~ @ A i . Therefore, there is no loss of generality if we

i~I

assume that B= ~ @ A j and B'= ~ @ A k where g = I - J . Let nl denote the na- j~J k~K

tural projection of M onto A~. We first consider the special case where A j , A i holds for all j6J and all

i=t=j. We claim that for such i and j, zcif(Af)c RadAj. Suppose this is not true.

Then njf(Ai)=Aj. Since Aj is A~-projective, the sequence A ~ A j - ~ O splits.

As Af is indecomposable, we obtain Aj~-A~, a contradiction. Therefore, for all kEK

f(Ak) ~ ~ ~j f(Ak) ~ ~ Rad Aj c Rad B. j~J j~J

Hence

Thus

f(B') = ~ f(Ak) ~ Rad B = B. kEK s

B =f(B| +f(B')=f(B),

and consequently B = y ' f (A) . Next we prove that this sum is direct. Given t~J, we have J~J

A t = 7~ t e = gt ( 2 f (A)) = ~t f ( A t ) " j sJ

As A n is d-continuous, gtflat is an isomorphism. Then it follows by Lemma 2.1 that

B =f(A, )O Ker nt[ B =f(A,) O ~,@ Aj. j~:t

Let F be a finite subset of J. Since each Ai is cyclic, ~ f ( A ) is contained in a jEF

finite direct sum ~ @ A j c B . Let N = ~ @ A j . Then N is d-continuous by j~F' jeF'

230 S. Mohamed and B.J. Miilter

Theorem 2.2. Now Lemma 2.3 applied to the d-continuous module N yields that ~ f ( A ) is direct. Since this is true for every finite subset F of J, we obtain

j e f

B = Z @ f ( A ) . j s J

Therefore f[~ is an isomorphism onto B. Again by Lemma 2.1, we get

M=B@Ker f .

Hence f : M--,B splits. Next we consider the general case. Let J ' be the subset of J consisting of all

j such that Aj~A , for all i=~j. Write B = C ' | where C'= ~ @ A j and j eJ"

C"= ~ @A i. Then for everyj6J' , A i is isomorphic to s o m e A i where i~j . Hence jr

Aj is Aj.-projective for all jr Therefore Aj is A,-projective for all jCa" and all ieI. Then by Theorem (G), Aj is (~@A3-projective, that is Ai is }/-projective

i e i

for all j6J'. Hence C" is M-projective by Theorem (E). Let n' and rd' denote the natural projections of B onto C' and C" respec-

tively. Since M-Z~+ C'--~ 0 is exact, we have, by the special case proved above,

M = C'@Ker n,f.

It is easy to see that f (Kern ' f )= C". Let f * = f l K e r ~ ' f , and consider the diagram

C"

Ker n ' f f* "" -~ C . . . . . . -~ 0.

Since C" is M-projective, C" is Kern'f-proiecfive, Hence there exists a homomorphism g: C " ~ K e r n ' f such that f ' g = 1. Therefore

Ker r( f =g( C")| Ker f *

= g(C")@KerJl Consequently

This completes the proof.

M = C'|

Corollary 2.5. Let M= ~@A~ where each A i is d-continuous and Aj-projective i e I

for all j 4 ~ i. I f Rad M ~ M, then M is d-continuous, s

Pro@ Since Ai is a summand of M, A s has small radical. By Theorem 2.4, A~ = ~ @Bj,, where each Bj~ is local d-continuous and Bk -projective for all k~EJ~,

lq4=jv Let r,s~I such that r~s. Since A~ is A~-projective, Bj,. is BjTprojective for all j~ j~. Therefore, we may write


M= 2 0 ;t~A

where C~ is local d-continuous and C,-projective for all /~=t=2. Then M is d- continuous by Theorem 2.4.

3. Applications

As a first application of Theorem 2.4, we study d-continuous modules with small radical over a local ring.

Theorem 3.1. Let M be a decomposable module over a local ring R. Then M is d- continuous with small radical if and only if M"~ @ R/I, where I is an ideal of R

; teA

and ]A[ < co if R/I is not right perfect. Moreover such a module M is quasi-projective.

Proof. Assume that M is d-continuous with small radical. Then by Theorem 2.4, we can write M = ~ @ Ax where A;t is local and Au-projective for all # + 2,

)~EA

Since M is decomposable, IAt > t. Now each A~ is isomorphic to R/B;t where B;~ is a right ideal of R. Let a, f leA and ~:~fl. Since B~ and B~ are small in R R and R/B/~R/B~ is d-continuous, R/B:~-R/B~ by ([6], Proposition 5.1). Hence R/B: is quasi-projective by Theorem (B). Since R is a projective cover of RIBs, B~ is an ideal of R (cf. [10], Proposition 2.2). Similarly B~ is an ideal of R. Then R/B~"~R/B~ implies that B~=B~. Let B~=I. Then B;t=I for every ,~eA.

Let R = R / I and consider M as an /~-module. It is obvious that M~ has small radical. Then, Lemma 28.3 of [-1] yields that R is right perfect.

Conversely, let M - @ R/I where t and A satisfy the conditions mentioned ;t~A

in the theorem. Since R/I is quasi-projective and R is local, R/I is d-continuous by ([6], Theorem 2.3). If A is finite, then M has small radical. On the other hand, if A is infinite, then R = R / I is a right perfect ring, consequently M~ has small radical, and therefore M R has small radical. Then M is d-continuous by Theorem 2.4.

The last statement is obvious.

Remark. Let M be an indecomposable module with small radical over a local ring R. Then it is easy to check that M is d-continuous if and only if M ~-RIB for some right ideal B such that for every unit x~R, x B ~ B implies x B = B .

As the last implication is true for every right ideal in a local right noe- therian ring R, all cyclic right modules over such a ring are d-continuous.

As a second application we determine the structure of all d-continuous modules with small radical over a commutative ring. (This result was suggested to us by a beautiful theorem of ZSschinger [11] concerning co-atomic modules.)

Theorem 3.2. Let R be a commutative ring. Then an R-module M is d-continuous with small radical if and only if M ~- ~,@ (R/I~) (~~ where the summation is tak- en over all maximal ideals m of R, where In, is an ideal of R contained in m and in no other maximal ideal, and where the cardinal number o3 is finite whenever R/I m is not perfect.

232 S. Mohamed and B.J. Mtiller

Proof First of all, let A and B be ideals of R contained in the maximal ideals m and n respectively, and in no other maximal ideals. Using that R is commutative, that A and B are comaximal if m q= n, and that R/A and R/B are modules over the local ring R/(Ac~B) if h i=n , it is easily seen that R/A and RIB are projective with respect to each other if and only if m + n or A =B.

If M is d-continuous with small radical, then we can write M = ~@A~ where A~ is local d-continuous and Ap-projective for all f i+e . Consequently A ~ R / I ~ for an ideal I~ contained in precisely one maximal ideal m~. From the mutual projectivity and the preceding observation, we obtain I~= I~ whenever m~=m~, and therefore M-~ ~ @ (R/I"*) (~"'). For infinite cardinal oo~,, Theo- rem 3.1 applied to the local ring R/I"*, shows that R/I., is perfect.

Conversely, let M = ~ @ (R/I.,) (~"*) be given, and let co"* be finite whenever R/I m is not perfect. In order to apply Theorem 2.4 to conclude that M is d- continuous we have to verify that R a d M is small. Therefore, suppose Rad M + K = M. We localize at the maximal ideal n, observing that (R/I,), = R/I, and that (R/I"*), = 0 if m ~ n since I m and rt are comaximal. We obtain

(n/I~) (9"~ + K~ = (R/I.) ('~'0.

Then Theorem 3.1, applied to the module M,~=(R/I.) (~ over the local ring R/I~, implies K,,=M.. Since this is true for all n, we obtain K=M. This completes the proof.

References

1. Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules. Graduate Texts in Mathe- matics 13. New York-Heidelberg-Berlin: Springer 1974

2. Azumaya, G.: M-projective and M-injective modules (1974, unpublished) 3. Azumaya, G., Mbuntum, F,, Varadarajan, K.: On M-projective and M-injective modules. Pa-

cific J. Math. 59, 9-16 (1975) 4. Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans.

Amer. Math. Soc. 95, 466-488 (1960) 5. Mohamed, S., Miiller, B.J.: Decomposition of dual continuous modules. In: Module Theory.

Proceedings (Seattle 1977), pp. 87-94. Lecture Notes in Mathematics 700. Berlin-Heidelberg- New York: Springer 1979

6. Mohamed, S., Singh, S.: Generalizations of decomposition theorems known over perfect rings. J. Austral. Math. Soc. 24, 496-510 (1977)

7. de Robert, E.: Projectifs et injectifs relatifs. C.R. Acad. Sci. Paris Ser. A 268, 361-364 (1969) 8. Singh, S.: Dual continuous modules over Dedekind domains. J. Univ. Kuwait Sci. (to appear) 9. Warfield, R.B.: A Krull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc.

22, 460-465 (1969) 10. Wu, L.E.T., Jans, J.E: On quasi projectives, Illinois J. Math. 11, 439-448 (1967) 11. Z6schinger, H.: Koatomare Moduln. Math. Z. 170, 221-232 (1980)

Received November 3, 1980; received in final form April 1, 1981

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Direct sums of dual continuous modules