A ringR is called right (m, n)-small injective if the rightR moduleRR is (m, n)-small injective

TRUONG CONG QUYNH

Abstract. LetR be a ring. Letm and nbe positive integers, a right R- moduleM is called (m, n)- small injective, if every right R-homomorphism from ann-generated submodule ofJ^mtoM extends to one fromR^mtoM.

A ringR is called right (m, n)-small injective if the rightR moduleRR is (m, n)-small injective. In this paper, we give some properties of (m, n)-small injective modules and right (m, n)-small injective rings.

Mathematics Subject Classification:16D50, 16D70, 16D80

Keywords:(m, n)-small injective module, (m, n)-small injective ring.

1. Introduction.

Throughout the paper,R represents an associative ring with identity 16= 0 and all modules are unitary R-modules. We write M_R (resp. _RM) to denote that M is a right (resp. left)R-module. Unless otherwise mentioned, by a module we will mean a rightR-module.

We recall some concepts and notations which will be used in this paper. We denote the Jacobson radical of a ring (resp. module) R (resp. M) by J (resp.

Rad(M)) and the injective hull of M by E(M). If A is a submodule of M, we denote byA≤M.

We write N ≤^e M, N M to indicate that N is an essential submodule, a small submodule ofM, respectively. A moduleM is called uniform ifM 6= 0 and every non-zero submodule of M is essential in M. A module M is calledto have finite uniform dimension, ifM does not contain an infinite direct sum of non-zero submodules. Recall that a moduleM is calledtorsionless, if given 06=m∈M, there existsα∈Hom(M, R) such thatα(m)6= 0, equivalently ifM can be embedded in a direct product of copies ofR. A ringRis calledright Kaschif every simple right R-module embeds in RR. A ring R is called semiregular if R/J is von Neumann regular and idempotents can be lifted modulo J. Note that if R is semiregular,

The work was supported by National Foundation for Science and Technology Development of Vietnam, project number, 101.01.29.09.

1

then for every finitely generated right idealI ofR,R=H⊕K, whereH ≤I and I∩KR.

A rightR-moduleM is called (m, n)-injective, if everyR-homomorphism from an n-generated submodule of R^m to M extends to one from R^m to M. In [2], some characterizations (m, n)-injective modules are given. It is proved that R is right (m, n)-injective (i.e. the right R-moduleRR is (m, n)-injective) if and only if everyRN in an exact sequenceRR^m→RRⁿ→RN→0 is torsionless. This result is similar to the Jain’s result - a ring R is right FP-injective if and only if every finitely presented rightR-module is torsionless (see [6]). A rightR-moduleMR is calledsmall injective, if every homomorphism from a right small ideal toMRcan be extended to aR-homomorphism fromR_R toM_Rand a ringRis called right small injective, ifRR is small injective. Yousif and Zhou introduced small injective rings (modules)(see [11]). They proved that a semiperfect ringRwith an essential right socle is right self-injective if and only ifR is right small injective. From this, some characterizations of QF rings in terms of small injectivity were obtained. Later, in [8], Shen and Chen claimed that if R is semilocal, then R is right self-injective if and only if R is right small injective. Under the small injectivity condition, they gave some new characterizations of QF rings and PF rings.

General background material can be found in [1], [3] and [10].

In this paper, we use the notationR^m×n for the set of allm×nmatrices overR.

ForA∈R^m×n,A^T will denote the transpose of A. In general, for anR-moduleN, we writeN^m×n for the set of all formalm×nmatrices whose entries are elements ofN. IfX ⊆M^l×m, S⊆R^m×n andY ⊆N^n×k, define

l_Ml×m(S) ={u∈M^l×m|us= 0,∀s∈S}

r_N^n×k(S) ={v∈N^n×k|sv= 0;∀s∈S}

r_Rm×n(X) ={r∈R^m×n|xr= 0,∀x∈X}

l_Rm×n(Y) ={r∈R^m×n|rs= 0,∀s∈Y} We will writeNⁿ=N^1×n, N_n =N^n×1.

2. Main Results.

Definition 2.1. A right R-module M is called (m, n)-small injective, if every R-homomorphism from an n-generated submodule of J^m (or J_m) to M can be extended to one from R^m(or Rm) to M. A ring R is called right (m, n)-small injective, ifRR is (m, n)-small injective.

Example 2.2. i)ZZ is (m, n)-small injective as aZZ-module, but it is not (m, n)- injective.

ii) LetR = { n x 0 n

!

| n ∈ ZZ, x ∈ZZ2} (see [11, Example 1.6]). Then R is a

commutative ring andJ =Sr={ 0 x 0 0

!

|x∈ZZ2}. ThereforeRis small injective but not self-injective. ThusR is (m, n)-small injective for all m and n. But it is easy to see thatR is not (1,1)-injective.

iii) Let R = F[x1, x2, ...], where F is a field and xi are commuting indeter- minants satisfying the relations: x³_i = 0 for alli, x_ix_j = 0 for all i 6= j, and x²_i = x²_j for alliandj. Then R is a commutative, FP-injective, local ring. We haveRis (1, n)-injective, butRis not a self-injective ring (see [7, Example 5.45]).

ThereforeR is (m, n)-injective for allmandnbut Ris not small injective.

We next consider some properties of (m, n)-small injective modules. By an ar- gument similar to the one given in the proof of [2, Theorem 2.4], we have:

Proposition 2.3. The following statements are equivalent for a right R-module M:

(1) M is(m, n)-small injective;

(2) lMⁿrR_n(α1, α2, . . . , αm) = M α1+M α2+· · ·+M αm for any m-element subset{α1, α2, . . . , αm} ofJⁿ.

The following result is a slight modification of [2, Theorem 2.9].

Proposition 2.4. The following statements are equivalent for a right R-module M:

(1) M is(m, n)-small injective;

(2) M is(m,1)-small injective and l_Mn(I∩K) =l_Mn(I) +l_Mn(K), where I andK are submodules of(Jm)R such that I+K is n-generated;

(3) M is(m,1)-small injective andlMⁿ(I∩K) =lMⁿ(I)+lMⁿ(K), whereIand Kare submodules of(J_m)_R such thatIis cyclic andKis(n−1)-generated (ifn= 1, thenK= 0).

The next characterization of (m, n)-small injective module is motivated by [2, Theorem 2.15]. One can prove it by Proposition 2.3 and an similar argument in the proof of [2, Theorem 2.15].

Proposition 2.5. The following statements are equivalent for a moduleM_R: (1) M is(m, n)-small injective;

(2) If m= (m₁, m₂, . . . , m_n)∈Mⁿ andA ∈J^m×n satisfy r_Rn(A)≤r_Rn(m), thenm=yAfor somey∈M^m.

Corollary 2.6. A right R-module M is (m, n)-small injective if and only if for everyA∈J^m×n,l_Mnr_Rn(A) =M^mA.

Corollary 2.7. Let R be a right (m, n)-small injective ring and M be a left R- module. IfR^m→Jⁿ→_RM →0 is exact, thenM is torsionless.

Proof. Let R^m →^f Jⁿ →R M → 0 be exact. There exists A ∈ J^m×n such that f(z) = zA for all z ∈ R^m and so Imf = R^mA. We claim that Jⁿ/R^mA is torsionless. In fact, let 0 6= ¯z ∈ Jⁿ/R^mA, where z = (z₁, z₂, ..., z_n) ∈ Jⁿ. By Proposition 2.5,rRⁿ(A)6⊆rRⁿ(z). Therefore there existsa= (a1, a2, ..., an)^T ∈Jn

such thatAa = 0 butza6= 0. Defineg : Jⁿ/R^mA →R such that g(¯x) =xa for every x∈Jⁿ. Clearly, g is well-defined, and g(¯z) =za6= 0. SoM 'Jⁿ/R^mA is torsionless.

Motivated by [7, Lemma 5.1], we give the following characterization of right (m, n)-small injective ring.

Theorem 2.8. The following statements are equivalent for a ring R:

(1) R is right(m, n)-small injective;

(2) l_Rn(BR_n∩r_Rn(A)) =l_Rn(B) +R^mA for allA∈J^m×n and B∈R^n×n; (3) IfrR_n(A)≤rR_n(B)withA∈J^m×n andB∈R^m×n, thenR^mB≤R^mA.

Proof. (1)⇒(2).Letx∈l_Rn(BR_n∩r_Rn(A)). For ally∈r_Rn(AB),ABy= 0 and so By ∈BRn∩rRn(A). It implies that xBy = 0 and soy ∈rRn(xB). Therefore rR_n(AB)≤rR_n(xB) orxB∈lRⁿrR_n(xB)≤lRⁿrR_n(AB). Since A∈J^m×n, then AB∈J^m×n. By Proposition 2.5, there existsy∈R^msuch thatxB=yAB. Thus x= (x−yA) +yA ∈ l_Rn(B) +R^mA. From this, we have l_Rn(BR_n∩r_Rn(A)) = lRⁿ(B) +R^mA.

(2) ⇒(1). Let B =In (identity matrix), then lRⁿrR_n(A) = R^mA. Thus R is right (m, n)-small injective by Corollary 2.6.

(1)⇒(3).Assume thatrR_n(A)≤rR_n(B). For eachx∈R^m, we haverR_n(B)≤ rR_n(xB) and sorR_n(A)≤rR_n(xB). It implies thatxB∈lRⁿrR_n(xB)≤lRⁿrR_n(A).

By Corollary 2.6,l_Rnr_Rn(A) =R^mAand hence xB ∈R^mA for all x∈Rⁿ. Thus R^mB≤R^mA.

(3)⇒(1).LetA∈J^m×n. We haveR^mA≤l_Rnr_Rn(A). For eachx∈l_Rnr_Rn(A), thenrRn(A)≤rRn(x). Let B = x

0

!

∈R^m×n. Therefore rRn(x) =rRn(B) and

hence rR_n(A) ≤ rR_n(B). By (3), R^m x 0

!

= R^mB ≤ R^mA. It follows that x∈R^mA and soR^mA=lRⁿrR_n(A). Thus Ris right (m, n)-small injective.

Proposition 2.9. The following statements are equivalent for moduleM: (1) M is(m, n)-small injective.

(2) For every n-generated submoduleIofJ^mand anyf ∈Hom(I, M), if(g, h) is the pushout of (f, i)in the following diagram (with iis the inclusion)

f

I R^m

M P

? -

i

?

g

-

h

there existsα∈Hom(P, M)such that αh=idM.

Proof. Similar to [12, Theorem 2.5].

The dual module ofP is denoted byP^∗=Hom(P, R).

Proposition 2.10. The following conditions are equivalent for a ring R:

(1) R is right(m, n)-small injective;

(2) If I is a m-generated and small submodule of a n-generated projective left R-moduleP, thenI=l_Pr_P^∗(I).

Proof. (1) ⇒ (2). Let I = Ra₁+Ra₂+· · ·+Ra_m be a m-generated and small submodule of a n-generated projective left R-module P. Since _RP is projective, there exist x1, x2, ..., xn ∈ P and f1, f2, ..., fn ∈ P^∗ such that x=

n

P

i=1

f(x)xi for allx∈ P. For eachx∈ l_Pr_P^∗(I), we have x=

n

P

i=1

f_i(x)x_i and a_j =

n

P

i=1

f_i(a_j)x_i for each j = 1,2, ..., m. Note that ai ∈ Rad(P), then f(ai) ∈ J for each i = 1,2, ..., m. Let αi = (fi(a1), fi(a2), ..., fi(am)), then αi ∈J^m, ∀i= 1,2, ..., n. Let ϕ:α₁R+α₂R+· · ·α_nR →Rviaϕ(

n

P

i=1

α_ir_i) =

n

P

i=1

f_i(x)r_i. It is easy to see that ϕis a homomorphism. By the hypothesis,ϕcan be extended toR^m. There exists u= (u1, u2, . . . , um)∈R^m such thatfi(x) =ϕ(αi) =uα^T_i =

n

P

j=1

ujfi(aj) for each i= 1,2, ..., n. Thusx=

n

P

i=1

fi(x)xi=

m

P

j=1

ujaj∈I.It implies that I=lPrP^∗(I).

(2)⇒(1).For eachα₁, α₂, . . . α_m∈Jⁿ. LetI=Rα₁+Rα₂+· · ·+Rα_m≤Rⁿ. By (2), I = lRⁿr(Rⁿ)^∗(I). But (Rⁿ)^∗ = Rn and so I = lRⁿrRn(I). Therefore Rα1+Rα2+· · ·+Rαm = lRⁿrR_n(α1, α2, . . . , αm). Thus R is right (m, n)-small

injective by Proposition 2.3.

Proposition 2.11. The following statements are equivalent for a ring R:

(1) Every n-generated right ideal ofJ^m is projective;

(2) Every quotient module of a (m, n)-small injective module is (m, n)- small injective;

(3) Every quotient module of a(m, n)-injective module is (m, n)- small injec- tive;

(4) Every quotient module of a small injective module is(m, n)- small injective;

(5) Every quotient module of an injective module is(m, n)-small injective.

Proof. Similar to [9, Theorem 2.17].

Next, we consider a case when the class of (m, n)-small injective modules coin- cides with that of (m, n)-injective modules.

Theorem 2.12. Let R be a semiregular ring. ThenM is(m, n)-small injective if and only ifM is(m, n)-injective.

Proof. Letf :K −→M_R be a R-homomorphism, where K is a n-generated sub- module of R^m. Since R is semiregular, then R^m is too. There exists a decom- position R^m = H ⊕L, where H ≤ K and K ∩L K. Hence R^m = K+L, K =H ⊕(K∩L) and so K∩L is a n-generated submodule of J^m. Thus there exists a homomorphismg: (R^m)R−→M such thatg(x) =f(x) for allx∈K∩L.

We construct a homomorphism ϕ: (R^m)R −→ M defined by ϕ(r) =f(k) +g(l) for anyr =k+l, k ∈K, l∈ L. Now we show thatϕis well defined. Indeed, if k₁+l₁ =k₂+l₂, wherek_i∈K, l_i∈L, i= 1,2, thenk₁−k₂ =l₁−l₂ ∈K∩L.

Hencef(k1−k2) =g(l1−l2), which implies that ϕ(k1+l1) =ϕ(k2+l2).Thusϕ

is a homomorphism andϕ_|K =f.

Corollary 2.13. Let R be a semiregular ring. ThenRis right(m, n)-small injective if and only ifR is right (m, n)-injective.

Note that a ringRis right (m, n)-small injective for all positive integersmand nif and only ifR is right (J, R)-FP-injective in the sense of Yousif and Zhou [11].

We shall conclude this paper with some properties of such rings.

Theorem 2.14. The following statements are equivalent for a ringR:

(1) R is right(m, n)-small injective for allm, n∈IN.

(2) R^n×n is right (1,1)-small injective for all n∈IN.

Proof. The result follows by [11, Lemma 1.3].

A moduleMR is FP-injective, if for every finitely generated submodule K of a free rightR-moduleF, every homomorphism fromKtoM extends to one fromF toM. In [7, Theorem 5.39], they proved thatRis right FP-injective if and only if Ris right (m, n)-injective for allm, n∈IN

From Theorem 2.12 and Theorem 2.14, we have:

Corollary 2.15. LetR be a semiregular ring. Then Ris right FP-injective if and only ifR is right(J, R)-FP-injective.

Proposition 2.16. IfRis right Kasch and right(J, R)-FP-injective, thenRis left (J, R)-FP-injective.

Proof. By Theorem 2.14, we claim that R^n×n is left (1,1)-small injective for all m ∈ IN. Since R is right Kasch, R^n×n is too. Let T = R^n×n. For each x ∈ J^n×n = J(T). Let y ∈ rTlT(xT) we need to show that y ∈ xT. Assume that y 6∈xT. LetL be a maximal submodule of xT +yT such that xT ≤L. Since R is right Kasch, there exists a T-monomorphism ϕ: (xT +yT)/L→T. Note that rTlT(J(T)) = J(T) and so y ∈J(T). Letψ :xT +yT → T viaψ(z) =ϕ(z+L) for all z ∈ xT +yT. By hypothesis, there is u ∈ T such that ψ = u·. Then ψ(y) =uy 6= 0 and ψ(x) = ux = 0 and so u∈ l_T(xT). But y ∈ r_Tl_T(xT) and henceuy= 0, this is a contradiction. ThusrTlT(xT) =xT. Acknowledgment. The author is deeply indebted to his supervisor, Professor Le Van Thuyet for his constant encouragement and guidance, and the referee for the very valuable comments and suggestions in shaping the paper into its present form.

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Truong Cong Quynh, Department of Mathematics, Danang University, 459 Ton Duc Thang, DaNang city, Vietnam

E-mail address:[email protected], [email protected]