Remarks on quasi-injective modules

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)

Ro g e r Yu e Chi Min g

(Paris VII, France)

Remarks on quasi-injective modules

Abstract. Semi-simple Artinian rings are characterized in terms of certain quasi-injective and continuous modules. A theorem of C. Faith and Y. Utumi on quasi-injective modules is shown to hold for a larger class of modules.

Introduction. Throughout, A represents an associative ring with identity and Л-modules are unitary. Z , J will denote respectively the left singular ideal and the Jacobson radical of A . A left A -module M is called p-injective (resp. Up- injective) if, for any principal (resp. complement) left ideal I of A, a e A, any left Л-homomorphism g: la -> M, there exists y e M such that g(ba) = bay for all b e l . Then A is von Neumann regular (resp. left continuous regular) iff every left Л-module is p-injective (resp. Lp-injective). It is well known that Л is von Neumann regular iff every left (right) Л-module is flat. Recall that a left Л- module M is quasi-injective if, for any submodule N of AM, any left Л- homomorphism from N into M may be extended to an endomorphism of AM.

Note that faithful quasi-injective projective left Л-modules need not be injective.

Following [3], a left Л -module M is called semi-simple iff J(M), the intersection of all maximal left submodules of M, is zero. Thus Л is semi-simple iff J = 0. As usual, an ideal of Л means a two-sided ideal.

Y. Utumi’s definition of left continuous rings [4] may be reformulated as follows: Л is a left continuous ring iff every left ideal of Л isomorphic to a complement left ideal of Л is a direct summand of A A . The notion of continuous rings has been extended to modules by S. Mohamed and T. Bouhy (cf. [2], P.

197). Left continuous modules may be redefined as follows:

De f in i t i o n.

A left Л-module M is called continuous if every submodule of AM isomorphic to a complement submodule of AM is a direct summand of AM.

We shall consider later a generalization of left continuous modules. Let us first characterize semi-simple Artinian rings in terms of quasi-injectivity, continuous modules and complement submodules. Recall that for any submodule N of AM , the closure of AN in AM is ClM(iV) = {y e M / ( N : y) is essential in

A\ [5]. Q denotes the maximal left quotient ring of Л whenever Z = 0. Write “Л is MELT” if every maximal essential left ideal of Л is an ideal.

Th e o r e m

1. The following conditions are equivalent :

(1) A is semi-simple Artinian;

(2) Every semi-simple left A-module is quasi-injective and flat ;

(3) A is a semi-prime ring whose faithful left ideals are quasi-injective and projective ;

(4) Every faithful left ideal o f A is injective and every simple right A-module is f l a t ;

(5) Every finitely generated faithful left A-module is quasi-injective;

(6) A is a left hereditary left Artinian ring with AQflat such that any left ideal o f A which is isomorphic to a complement left ideal is a complement left ideal;

(7) A is a left continuous regular ring such that for any cyclic left A-module M and any non-zero submodule N isomorphic to a complement submodule of AM, N — C\M(N);

(8) Every faithful left A-module is continuous;

(9) A is a MELT ring whose cyclic semi-simple left modules are continuous and flat.

P ro o f. (1) implies (2) and (3) obviously.

Assume (2). Since А /J is a semi-simple left Л-module,

A/J is flat which yields J = 0. Therefore, A is left self-injective regular which implies that every left ideal is semi-simple, and hence quasi-injective. But then, every simple left A- module is injective by [6], Lemma 1.1, whence every left Л-module is semi­

simple [3], Theorem 2.1. Thus every left Л-module is quasi-injective and (2) implies (5).

Assume (3). Since Л is left self-injective, we have Z = J . If К is a complement left ideal such that L = J @ K is an essential left ideal, then Л semi­

prime and J = Z is an ideal of Л together imply that AL is faithful. Now AJ is projective (since AL is projective) and if J ф 0, we have J = the Jacobson radical of J — J 2 Ф J, a contradiction! Thus Z = J = 0, whence any essential left ideal of Л is faithful and therefore projective. This proves that Л is left self- injective left hereditary and (3) implies (4) by a result of B. Osofsky.

Assume (4). Suppose there exists b e A such that AbA + l(b) ф A. Let M be a maximal left ideal containing AbA + 1(b). If M is not an ideal of Л, then for any a e A , aM = 0 implies <яеаЛ = а М Л = 0 , whence AM is faithful. Then M = 1(e), e — e2eA . Now eer(l(b)) = r(l(bA)) = bA ç M which implies e2 = 0, a contradiction. Thus M = M A and M is also a maximal right ideal of Л. Now A /M

is flat which yields b = db for some d e M . Then 1 —d e 1(b) ç M which implies that 1 e M , again a contradiction. This proves that AbA + 1(b) = A for all b e A which yields Л fully left idempotent. Therefore Z = J = 0 and Л is left self-injective regular. It then follows that any essential left ideal L of Л is faithful whence L = A. This proves that (4) implies (5).

Assume (5). For any finitely generated left Л-module M, let В =

A ®

M .

Then AB is a finitely generated faithful left Л-module which is therefore quasi-

injective. This implies that AM is quasi-injective which proves that every left Л-

module is injective and projective and therefore (5) implies (6).

Assume (6). Then Z = 0. For any left ideal I of A isomorphic to a complement left ideal, since / is a complement left ideal, then I = Cl^ (/) and

a

A/I is non-singular by [5], Theorem 4, which implies

A/I projective by [1], Theorem 5.21. Thus AI is a direct summand of aA which implies A left continuous regular and (6) implies (7).

Assume (7). If M is a cyclic left Л-module and N a submodule isomorphic to a complement submodule of AM, then N = C1M(N) implies that AM /N is non-singular [5], Theorem 4.

Since A is left continuous regular, every cyclic non-singular left Л-module is projective which proves that AN is a direct summand of AM. Thus any cyclic left Л-module is continuous which implies Л semi-perfect [2], p. 201. Since J = 0, then A is semi-simple Artinian and (7) implies (8).

Assume (8). For any left ideal / of Л, M =

I ® a A is a faithful left Л- module which is therefore continuous. By [2], Lemma 2.3, AI is a direct summand of aA which proves that (8) implies (9).

Finally, (9) implies (1) : Since

A/J is flat, we have J — 0 which implies that Л is left continuous regular. Then any prime factor ring of Л is a MELT left self- injective regular ring which is therefore Artinian. Л is then a left F-ring [6]

which implies that every cyclic left Л-module is semi-simple and hence continuous. Thus Л is semi-simple Artinian [2], p. 201.

Co r o l l a r y

1.1. Л is a finite direct sum of division rings iff A is a reduced ring whose faithful left ideals are quasi-injective and projective.

We now introduce the following generalization of left continuous Л- modules and quasi-injective left Л-modules.

De f in i t i o n.

A left Л-module M is called GQ-injective (generalized quasi- injective) if, for any submodule N of AM which is isomorphic to a complement submodule of AM, every left Л-homomorphism from N into M may be extended to an endomorphism of AM.

A is called a left GQ-injective ring if aA is GQ-injective. It is well known that continuous regular rings need not be self-injective [4], p. 172. Thus GQ- injectivity generalizes effectively quasi-injectivity even for commutative regular rings.

For any ring R, J ( R ) denotes the Jacobson radical of R. A theorem of C.

Faith and Y. Utumi [1], Theorem 2.16, asserts that if M is a quasi-injective left Л-module, E = End(AM), then E/J(E) is von Neumann regular and J (E )

= { f e E / k e r f is essential in AM}. We show that this result holds for GQ- injective modules.

Th e o r e m

2. Let M be a GQ-injective left A-module. I f E = End(^M), then (1) E/J (E) is von Neumann regular and (2) J (E) = { /e Е/ker/ is essential in AM } .

P ro o f. If we set U = { /e £ / k e r /i s essential in AM], then U is an ideal of

E. We show that U Я J{E). I f f e U , c e E , since k e r / n k e r ( l — cf) = 0, then

ker(l —cf) = 0. The isomorphism 1 —cf of M onto (1 —c f ) M yields an inverse

isomorphism g from (1 —c f ) M onto M. Since AM is GQ-injective, g may be extended to an endomorphism h e E such that h(l — cf) is the identity map on AM. This proves that 1 — с / is left invertible in E for any c e E which implies f e J ( E ) . Thus l/_ ç J(E).

(1) Let 0 Ф J

E/J(E), f e E . Then f $ U (since U ç= J (E)) and by Zorn’s Lemma, there exists a non-zero complement submodule К of AM such that ker f@ K is a left essential submodule of AM. If г: К -+ M is the restriction of / to K , then r is an isomorphism of К onto r(K) and there exists an isomorphism s of r(K ) onto К such that sr is the identity map on K. Since К is a complement submodule of AM if i: К -* M is the canonical injection, we have that: r(K ) -> M may be extended to t e E (AM being Gg-injective). For any k e K , t ( f (к))

= к which implies that fC + k e r / ^ ker (ft f—f ) , whence f t f —f e J { E ) . Thus/ г /

= / in E/J (E) which proves that E/J (E) is von Neumann regular.

(2) Suppose there exists v e J (E ) such that v$ \J. Then the proof of (1) shows that there exists z e E such that w = v — vzveU . Since there exists an inverse t of 1 — zv in E, we have v = v ( l —zv)t = w te U (an ideal of E), which is a contradiction. This proves that J (E) = U.

Since continuous left Л-modules are G()-injective, the next result then partially generalizes [4], Lemma 4.1.

Corollary

2.1. Let M be a continuous left A-module and E — EndA{M).

Then E/J(E) is von Neumann regular, where J (E) = {f e £/ker/ is essential in

a

M}.

Corollary 2.2.

I f A is a left p-injective ring such that every complement left ideal is principal, then А/J is von Neumann regular.

Corollary 2.3.

I f A is a semi-simple left GQ-injective ring, then A is von Neumann regular.

We exploit further the notion of faithfulness concerning modules and ideals.

Th eo rem 3.

The following conditions are equivalent : (1) A is left continuous regular;

(2)

A is a semi-prime ring such that any faithful left ideal is either Up-injective or a left annihilator;

(3)

A is a semi-simple left GQ-injective ring whose complement left ideals are finitely generated;

(4) Every faithful left A-module is Up-injective.

P ro o f. It is easy to see that (1) implies (2).

Assume (2). If К is a complement left ideal such that L = Z © К is an

essential left ideal, then AL is faithful since A is semi-prime and Z is an ideal of

A . If aL is Up-injective, then AZ is l/p-injective which implies Z = 0. In case L is

a left annihilator, we have L = A which again implies Z = 0. Now every

essential left ideal is a faithful left Л-module and since any left ideal is a direct

summand of some essential left ideal which is either I/p-injective or a left annihilator, any principal or complement left ideal is a direct summand of aA which proves that (2) implies (3).

(3) implies (4) by Corollary 2.3.

Since a direct summand of a l/p-injective left Л-module is C/p-injective, the proof of Theorem 1 shows that (4) implies (1). a

The next proposition may similarly be proved.

Pr o p o s it io n

4. Let A be a semi-prime ring. Then

(1) A is von Neumann regular iff every faithful left ideal is either p-injective or a left annihilator ;

(2) A is regular left hereditary iff A is a left p-injective ring such that every faithful left ideal is projective;

(3) A is a left hereditary principal left ideal ring iff every faithful left ideal of A is principal projective.

R e m a r k 1. A is von Neumann regular iff any one of the following conditions is satisfied : (1) Every finitely generated faithful left A -module is flat ; (2) Every finitely generated faithful left Л-module is p-injective ; (3) The sum of any two injective left or right Л -modules is p-injective and flat.

R e m a r k 2. If Л is a prime left non-singular ring whose complement left ideals are direct summands, then all the minimal left ideals of Л (if they exist) are isomorphic.

R e m a r k 3. Л is left self-injective regular iff Л is a left semi-hereditary ring with aQ flat and the multiplication map of QA®Q into Q is an isomorphism such that any left ideal of Л isomorphic to a complement left ideal is a complement left ideal. (Apply [1], Theorem 5.18.)

R e m a r k 4. Л left Gg-injective integral domain is a division ring.

References

[1] K. R. G o o d e a r l, Ring Theory : Non-singular rings and modules, Pure Appl. Maths. 33, Dekker (1976).

[2] S. K. J a in and S. M o h a m e d , Rings whose cyclic modules are continuous, J. Indian Math. Soc.

42 (1978), 197-202.

[3] G. O. M ic h le r and О. E. V i 11a m ay or, On rings whose simple modules are injective,i. Algebra 25 (1973), 185-201.

[4] Y. U t um i, On continuous rings and self-injective rings, Trans. Amer. Math. Soc. 118 (1965), 158—

173.

[5] R. Y u e C h i M in g , A note on singular ideals, Tôhoku Math. J. 21 (1969), 337-342.

[6] —, On regular rings and self-injective rings, Monatshefte ftlr Math. 91 (1981), 153-166.

UNIVERSITE PARIS VII U.E.R. DE MATHEMATIQUES PARIS, FRANCE