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Rings characterized by injectivity classesWang Dingguo a ba Institute of Mathematics , Fudan University , Shanghai, 200433b Qufu, Shandong, 273165, P.R ChinaPublished online: 27 Jun 2007.

To cite this article: Wang Dingguo (1996) Rings characterized by injectivity classes, Communications in Algebra,24:2, 717-726, DOI: 10.1080/00927879608825594

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COMMUNICATIONS IN ALGEBRA, 24(2), 717-726 (1996)

RINGS CHARACTERIZED BY INJECTIVITY CLASSES

Institute of Mathematics, Fudan University Shanghai 200433, and

Department of Mathematics, Qufu Normal University Qufu, Shandong 273165, P.R. China

ABSTRACT. Recently, there are a number of well known theorems which char- acterize rings in terms of injective modules (cf. [2], [3], [5] and [8]). The purpose of this paper is to give general theorems which show that some similar results remain true when many other classes of general injective modules are considered in place of injective modules. Our results encompass several well known results by Huynh and Smith, Liu, and Xue. In particular we answer negatively the question of Yue Chi Ming (111 and a n example is given to show tha t E-injective modules form a n intermediate class between continuous and quasi-continuous modules(cf. Example 3).

1 .INJECTIVITY CLASSES

Recall that (1) A left R-module M is NCI [Q] iff for any submodule P

containing a non-zero complement submodule of M and any submodule N of

M which is isomorphic to P, every R-homomorphism from N iAto P extends

to an endomorphism of M.

(2) M is SQC (101 iff for any submodule N of M such that there exist

a non-zero complement submodule C of M which is isomorph~c to a factor

module of N , then any R--homomorphism from N into M may be extended to

an endomorphism of M.

(3) M is E-injective [ll] iff for any non-zero complement submodule

C of M and relative complement K of C, any essential submodule E of M

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Copyr~ght @ 1996 by Marcel Dekker. Inc.

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containing K $ C, any R-monomorphism g : E -+ M and R-homomorphism

f : E -+ M, there exists an endomorphism h of R M such that hg = f .

(4) Consider the following conditions on a module RM:

( C l ) Every submodule of M is essential in a summand of M .

(C2) Every submodule isomorphic to a summand of M is itself a sum-

mand.

(C3) If MI and Mz are summands of M with MI n Mz = 0, then MI $Mz

is a summand of M.

M is called continuous if it satisfies conditions (Cl) and (C2), quasi-

continuous if it satisfies (Cl ) and (C3), and a CS-module if it satisfies condition

(Cl ) only.

It is easy to see that (C2) implies (C3), but the converse is not true

in general. Thus, every continuous module is quasi-continuous. The ring of

integers 2 is an example of a commutative, noetherian, quasi-continuous ring

which is not continuous (and hence not quasi-Frobenius). For a full account of

the subject of (quasi-)continuous modules and CS-modules we refer the reader

to [l] and [6].

Throughout, all rings considered have an identity and modules are unital

left modules and E(M) , J ( M ) , Soc(M) and Z(M) stand respectively for the

injective hull, the Jacobson radical, the socle and the singular submodule of

M. In particular, J = J ( R ) , Z = Z(R). We will freely make use the notation,

terminology and results of Wisbauer [7].

Let c be any cardinal. Following [2], an R-module M will be called t

c-limited provided every direct sum of non-zero submodules of M contains at

most c direct summands. M is called an ES-module if the socle Soc(M) is

an essential submodule of M (See [5]). For example, if M has finite uniform

dimension then M is No-limited and any R-module M is c-limited, where

c = /MI.

Now, let Z be a class of modules, that is a collection of R-modules such

that if M E Z then any R-module isomorphic to M belongs to 1. We consider

the following condition about a class of modules Z.

Definition We call a class Z of R-modules an injectivity class if it is closed

under direct summands, contains all self-injective (=quasi-injective) modules

and M $ E ( M ) E Z implies that M is injective.

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INJECTNITY CLASSES 719

Yue Chi Ming proved that S Q C modules are intermediate between quasi-

injective and quasi-continuous modules, N C I modules are continuous modules,

and E-injective modules to be an intermediate class of modules between con-

tinuous and CS modules (cf. [9, 10 and 111).

We will see that there exist many injectivity classes.

Examples 1. From [3], [4], [6], [8], [9] and [lo], we know that the class of all

quasi-injective modules, the class of all vN-injective (see [4] for the definition)

modules, the class of all (quasi-)continuous modules, the class of all direct-

injective modules, the class of all NCI-modules and the class of all SQC-modules

are all injectivity classes.

2. However, the class of all CS-modules is not an injectivity class. For

example, for any prime integer p, consider N = ZlpZ as Zlp2Z-module, then

E ( N ) = Zlp2Z. By [ I , p.591, we know that (ZIpZ) $ (Zlp2Z) is C S as a

Zlp2Z-module but Z l p Z is not Zlp2Z-injective.

The following proposition shows that the class of all E-injective modules

is an injectivity class.

Proposi t ion 1. For any R-module M I if M $ E ( M ) is E-injective, then M

is injective.

Proof. Let N = M $ E ( M ) . If j : M + E ( M ) , k : E ( M ) + N are the

inclusion maps, i : M + M the identity map, q : N --+ M and p : M $ C --+

M are the natural projections, where C is a relative complement of M in N ,

then k j p : M $ C + N extends to an endomorphism h of N, since N is

E-injective. Now i is the restriction of h to M I then g = qhfe is a map of

E ( M ) into M such that g j = i. This proves that M is a direct summand of

E ( M ) , whence M = E ( M ) is injective. 0

In 111, Proposition 21, it was shown that if R is left E-injective ring and

every complement left ideal of R is an ideal, then RIZ is von Neumann regular

and Z = J, and Yue Chi Ming also asks whether this result holds for arbitrary

left E-injective rings.

We note that the above result does not hold for arbitrary E-injective

rings. For by [12, Theorem 11, it is clear that E-injective modules are quasi-

continuous modules. From [6, Proposition 3.151, if for every left E-injective

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ring R, R/Z is von Neumann regular and 2 = J, then every E-injective

ring is continuous rings. But t h s is not the case since there exist semiprime

E-injective rings which are not continuous regular rings(cf.[ll, p.1401).

The following example shows that quasi-continuous (uniform) modules

need not be E-injective.

Example 3 The ring of integers Z is an example of a quasi-continuous ring

which is not E-injective since for the monomorphism f : Z + Z, f ( z ) = 22

and g : Z -+ Z,g(z ) = 2, there doesn't exist h : Z ---+ Z such that hf = g.

2. CHARACTERIZING RINGS BY INJECTIVITY CLASSES

In the following we assume that Z be an injectivity class of modules.

First we give a characterization of hereditary rings by means of the injectivity

class Z.

Theorem 2. The following conditions are equivalent for a ring R:

(a) R is hereditary.

(b) Every factor module of an injective R-module is in 2.

(c) Every sum of two injective submodules of an R-modules is in Z.

(d) Every sum of two isomorphic injective submodules of an R-module

is in Z.

Proof. The implication (a) + (b) and (c) ==+ (d) are trivial.

(b) * (a) Let I be an injective R-module and K a submodule. We shall

prove that I / K is injective. Let p : I + I / K be the natural epimorphism, then

we have an epimorphism E ( I / K ) $I -+ E ( I / K ) $ ( I I K ) . By (bf , E ( I / K ) $

( I I K ) is in Z. Then I / K is injective.

(b) + (c) Let I2 and I2 be two injective submodules of an R-module M.

Since Il $ I2 is injective and there is an epimorphism Il $ I2 -+ Il +I2, Il + Iz

is in Z by (b).

(d) 3 (6) Let I be an injective R-module and K a submodule. Let

U = I $ I , V = {(z ,z) i U I z i K ) , U = U/V,Il = {(i,O)u/i i I}, and

I2 = {(O, i) E 1 i i I}. Then u = II + I2 and I, I ( i = 1,2), so u is in Z - -

by (d). Since Il is a summand of U, U/I1 is isomorphic to a summand of u .

Hence u/Il is in Z. Now there is a canonical isomorphism I / K E u/Il via -

i + K -+ (0 , i ) + II and so I / K is in 2. 0

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INJECTIVITY CLASSES 72 1

Koehler [3] has characterized Artinian semisimple rings using quasi-

injective modules. We will be concerned with the following question: given

a ring R all of whose left R-modules are in 2, what implications does this have

for the ring R itself ? Now we give a nice characteristic properties of semi-simple

Artinian rings in terms of the injectivity class 2 .

T h e o r e m 3. The following conditions are equivalent for a ring R:

(1) R is left semisimple.

(2) 2 = R - Mod.

(3) R R E Z and 2 is closed under finite direct sums.

(4) A left R-module is flat iff it belongs to 2.

(5) R R is injective and 2 coincides with the class of all injective left

R-modules.

(6) R is a semi-prime ring whose faithful left modules are in 2.

(7) Any direct sum of a projective left R-module and a quasi-injective

or a left R-module in Z is in 2.

(8) A left R-module is projective iff it belongs to Z.

(9) Every left R-module in Z is projective.

Proof. The implication ( 1 ) * (2) =+ (3 ) , ( 1 ) =+ ( 8 ) , ( 1 ) + (4), (5)) (6), (7)and(9)

are trivial.

( 3 ) ( 1 ) If M E 2, then E ( M ) $ M E Z by hypothesis, whence M

is injective. We have shown that each R-module in Z is injective whence RR

in injective. Since quasi-injective modules are in 2, then the direct sum of I

two quasi-injective R-modules is injective. By [3, Corollary 2.4]', we get R is

Artinian semisimple.

( 4 ) ==+ ( 1 ) If M E Z , E ( M ) the injective hiill of M , then M and E ( M )

are flat which implies that M $ E ( M ) is flat. Therefore M $ E ( M ) E Z and

we get that M is injective. Then every projective left R-module is injective

whence R is quasi-Frobeniu by [7, 48.151.

Since every simple left R-module is in Z , then R is a left V-ring which

yields R is semiprime Artinian ring, i.e Artinian semisimple ring.

( 5 ) ==+ ( 1 ) Since every quasi-injective module is in 2. Then any direct

sum of two quasi-injective modules is quasi-injective, by (31, Corollary 2.4, R is

Artinian semisimple.

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(6) =+ (1) Let C be a complement left ideal of R such that A = Z $ C

is an essential left idel. If xA = 0, z E R, then (Az)' = 0 implies x E Z. Then

(Rx)' Z z Az = 0 implies z = 0, that is, RA is a faithful module. Thus

RA $ E ( R A ) is faithful and whence RA $ E(RA) is in Z, and we get that RA

is injective. Therefore R = A and Z = 0 since Z has no non-zero idempotent.

Now R is a semi-prime left non-singular ring. By the above argument, we get

any left essential ideal L = R. This prove that R is Artinian semisimple.

(7) a (1) For any projective R-module M , by (7), M $ E ( M ) E Z,

thus M is injective, that is, every projective left R-module is injective. Then

R is quasi-Frobeniusen (see [7, 48.151). For every simple left R-module S, the

injective hull E ( S ) is projective since R is a QF-ring, from this it follows that

S is injective, that is R is left V-ring which yields R is Artinian semisimple.

The proof of (8) e (1) analogous as that of (4) =+ (1).

(9) =+ (1) Since every simple left R-module is in Z, by condition (9),

we have that every simple left R-module is projective. Therefore R is Artinian

semisimple ring [7, 20.71. 0

Dinh van Huynh and P.F.Smith [2] characterized Artinian semisimple

rings using injective modules. Liu [5 ] characterized Artinian semisimple rings

using continuous modules. Using their ideas we are now in a position to give

a nice characteristic properties of semi-simple Artinian rings in terms of the

injectivity class Z. This result generalizes the corresponding results of Dinh

van Huynh and P.F.Smith [2] and Liu [5 ] . 1

Theorem 4. The following conditions are equivalent for a ring R:

(1) R is left semisimple.

(2) There exists a cardinal c such that every R-modules is the direct

sum of a module in Z and i c-limited ES-module.

Proof. The implication (1) ==+ (2) is trivial.

(2) =+ (1). Suppose that M is a left R-module. It is enough to show

that M is injective. Set N = M $ E ( M ) . Let {S, : w E R) denote a collection

of representatives of the isomorphism classes of simple left R-modules and let

S = ewEnS,. Let K be an index set with 1K1 2 c, and for each a E K,

let Ta = S, define T = @aEKT,. Let I be an index set with 111 > IE(T)(.

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INJECTIVITY CLASSES 723

For each x in I let N, = N , and F = By assumption, there exists

A E Z and a c-limited ES-module B such that F = A$ B. Note that Soc(B)

is a direct sum of at most c simple submodules of B, it is clear that there

exists a monomorphism f : Soc(B) --+ T. Thus we obtain a homomorphism

g : B --+ E(T) such that glS,,(B) = f . Since B is an ES-module, Soc(B) is

an essential submodule of B , which implies that g is a monomorphism. Thus

we have IBI < IE(T)I. For each b E B, there exists a finite subset I(b) of

I such that b E $IEI(b)N=. Let I' = ~ ~ & B I ( ~ ) . If IBI is finite, then ]It( is

finite. Thus 11'1 6 IE(T)I. Now suppose that (BI is an infinite cardinal, then

11'1 < IBI < IE(T)I. Set I" = I - I t . From the construction of I it follows that

111 > IE(T)I, and thus I" # 0. Now let G = $zEpN2,H = $,EIuN,. Then

we have F = G $ H = A $ B , and B < G. Thus it follows by modularity

that G = (A n G) $ B. So F = A $ B = (A n G) $ B $ H, which implies that

A r (AnG)$H. Since A E Z, it follows that H E Z, too. Thus N = M @ E ( M ) ,

a direct summand of H I is in 1. Hence M is injective and we are done.

Theorem 5. The following conditions are equivalent for a ring R:

(1) R is left Noetherian.

(2) There exists a cardinal c such that every direct sum of injective left

R-modules is the direct sum of a module in Z and a c-limited ES-module.

Proof. The implication (1) (2) is trivial by [2], Theorem 4.

Suppose that R satisfies (2). Let {S, : w E 0) denote a collection of

representatives of the isomorphism classes of simple left R-modules and let I

S = Let A be an index set with /A1 3 c, and for each X E A, let

Tx = S. Define 7' = aXEhTX and let k = IE(T)I.

Let Ui(l < i < m) be simple left R-modules and for each i 2 1, let

E, = E(U,). Let M = $i21E,. and N = M $ E(M) . Let I be an index set with

111 > k. For each x in I let N, = N, and F = @rEINz. By assumption, there

exists a R-module A E Z and a c-limited ES-module B such that F = A$B.

' By analogy with the proof of (2 ) + (1) in Theorem 4, we obtain that N E Z.

Thus M = $i21E, is injective. Thus every countable direct sum of injective

hulls of simple left R-modules is injective. By [7 , 27.31, R is left Noetherian.

0

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724 WANG

R is a QF-ring if and only if each injective left R-module is projective

if and only if each projective left R-module is injective( i.e.[7,48.15] Noetherian

QF rings).

Theorem 6. The following conditions are equivalent:

(1) R is a QF-ring;

(2) 2 contains the direct sum of projective and injective modules;

(3) There exists a cardinal c such that every direct sum of injective or

projective R-modules is the direct sum of a module in Z and a c-limited

ES-module.

Proof. (1) @ (3) may be proved along lines similiar to those used to prove

Corollary 11 in [5].

Since R is Q F if and only if every projective left R-module is injective

if and only if every injective left R-module is projective, then (1) implies (2).

Assume (2). For any projective left R-module M I then M $ E(M) E Z

whence M is injective. Thus R is Q F and hence (1) implies (2). 0

By the conditions on Z, we have seen that every (quasi-)injective module

is in Z. And, in general, Z is not closed under direct sums. Now we investigate

under what condition(s) Z is closed under direct sums or every module in Z is

injective ? We have the following results.

Proposition 7. The following conditions are equivalent:

(1) Z is closed under direct sums;

(2) R is a left QI-ring and each module in Z is injective; t

(3) R is a left Noetherian ring and each module in Z is injective;

Proof. The implication (2) e~ (3) and (2) (1) are trivial by the fact

that every quasi-injective is in Z and for Noetherian ring R every direct sum af

injective modules is injective. We can prove that every left R-module in Z is

injective. Thus R is a left QI-ring since every quasi-injective left R-module

is in 2. This proves (1) =+ (2). 0

The following is an immediate consequence of the previous proposition

and the fact that every simple left R-module is in Z.

Proposition 8. If Z is closed under finite direct sums, then R is a left Noe-

therian, left hereditary, left V-ring.

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INJECTMTY CLASSES

Proposition 9. The following conditions are equivalent:

(1) Any module in Z is injective.

(2) 2 is closed under finite direct sums.

(3) 2 is closed under any direct sums.

Proof. By Proposition 7, the implication (3) + (2) (1) are trivial.

Assume (1). Since any quasi-injective left R-module is in Z, therefore

R is a left QI-ring and hence (1) implies (3). 0

3. CHARACTERIZING MODULES BY INJECTIVITY CLASSES

Following Wisbauer[7], for any M E R-Mod, we denote by o[M] the full

subcategory of R-Mod, whose objects are the submodules of M-generated

modules. For N E u[M], EM(N) denotes the injective hull of N in o[M].

EM(N) is also called the M-injective hull of N and is isomorphic to the trace

of M in E(N). The object of the following is to m o w and make a straight-

forward extension of the definition of injectivity classes in R-Mod to the full

subcategory u[M]. In a similar vein we again get the analogous results for

modules.

Definition For an R-module M we call a class 2 of modules in u[M] an

injectivity class in the category u[M], if it is closed under direct summands,

contains all self-injective (=quasi-injective) modules in u[M] and N @ EM(N) E

Z implies that N is M-injective.

Definition Let M be an R-module. P E u[M] is called hereditary in u[M] if

every submodule of P is projective in u[M]. I

Using (7, 27.3 and 39.81, analogously as above proof, we can prove the

following results.

Theorem 10. If M is projective in a[M], then the following are equivalent:

(a) M is hereditary in a[M];

(b) Every factor module of an M-injective R-module in o[M] is in 1;

(c) Every sum of two M-injective submodules of an R-modulesin u[M]

is in 1;

(d) Every sum of two isomorphic M-injective submodules of an R-module

in o[M] is in Z.

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WANG

Theorem 11. For an R-module M the following are equivalent:

(a) M is semisimple;

(b) For some injectivity class 1, there exists a cardinal c such that every

module in u [ M ] is a direct sum of a module in Z and a c-limited ES-module.

Theorem 12. For an R-module M the following assertions are equivalent:

( I ) M is locally Noetherian;

(2) There exists a cardinal c such that every direct sum of M -injective

left R-modules in a [ M ] is the direct sum of a module in Z and a c-limited

ES-module.

ACKNOWLEDGEMENT

The author is grateful to Professor Xue Weimin and R.Wisbauer for

many helpful comments and suggestions and he is also grateful to Professor

R.Yue Chi Ming for providing reprints of [9], [lo] and [ l l ] .

REFERENCES

[I] N.V. Dung, D. Van Huynh, P.F. Smith and R. Wisbauer, Eztending modules, Pitman Research Notes in Math. Ser. 313, Longman Sci. and Tech. 1994.

[2] Dinh van Huynh and P.F. Smith, Some rings characteriaed by their modules, Comm. in Alg. 18(6) (1990), 1971-1988.

(31 A. Koehler, Quasi-projective covers and direct sums, Proc. Amer. Math. Soc. 24(4) (1970), 655-658.

[4] M.S. Li and J.M. Zelmanowitz, O n the generalizations of injectivity, Comm. in Alg. 16(3) (1988), 483-491.

[5] Liu Zhongkui, Characterizations of rings by t h k modules, Comm. in Alg. 21(10) (1993), 3663-3671.

[6] S.H. Mohamed and B.J. Miiller, Continuous and discrete modules, Londoh Math. Soc. Lecture Note Series 147 (Cambridge University Press 1990).

(71 R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.

(81 Xue Weimin, Characierazation of rings using direct-projective modules and direct-injective modules, J . of Pure and Appl. Alg. 87(1) (1993), 99-104.

(91 R. Yue Chi Ming, On von Neumann regular rings, X , Collectanea Math. Sem. Mat. Barcelona 34 (1983), 81-94.

[lo] R. Yue Chi Ming, On quasi-injeciivity and quasi-continuity, Serdica 0 (1983), 301-306. [ I l l R. Yue Chi Ming, On generalizations of self-injective and strongly regular rings, Kyung-

pook Math. J . 28(2) (1986), 137-148. [12] R. Yue Chi Ming, On generalizations of injectivity, Arch. Math.(Brno) 28 (1992), 215-

220.

Received: March 1995

Final revised version: August 1995

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