Rings Characterized by Projectivity Classes

International Journal of Algebra, Vol. 4, 2010, no. 19, 945 - 952

Rings Characterized by Projectivity Classes Yongduo Wang and Dejun Wu Department of Applied Mathematics Lanzhou University of Technology Lanzhou 730050, P. R. China [email protected], [email protected]

Abstract. Some characterizations of δ-semiregular (semiperfect, perfect, semisimple) rings are given. Our results generalize several well known results. Mathematics Subject Classification: 16E50, 16l30 Keywords: δ-small submodule, δ-semiregular ring, δ-(semi)perfect ring, semisimple ring, projectivity class

1. Introduction and Preliminaries As generalizations of semiregular rings and (semi-)perfect rings, the notions of δ-semiregular rings and δ-(semi-)perfect rings were introduced by Zhou [14]. Wang [9] gave characterizations of semiregular rings and (semi-)perfect rings by introducing the concept of projectivity classes of modules. Motivated by this, we will characterize δ-semiregular rings and δ-(semi-)perfect rings in term of projectivity classes of modules in section 2. It is well known that [7] a ring R is a semiregular ring if and only if RR is an amply finitely supplemented module if and only if RR is a finitely supplemented module. We prove that a ring R is a δ-semiregular ring if and only if RR is an amply finitely δ-supplemented module if and only if RR is a finitely δ-supplemented module in section 3. In this note all rings are associative with identity and all modules are unital left modules unless specified otherwise. Let R be a ring and M a module. N ≤ M will mean N is a submodule of M. A submodule N of M is said to be δ-small in M [14], denoted by N δ M, if whenever N + X = M with M/X singular, we have X = M. Every small submodule or non-singular semisimple submodule of M is δ-small in M. Let ℘ be the class of all singular simple modules. Following [14], δ(M) = RejM (℘) = {N ≤ M | M/N ∈ ℘} is the reject in M of ℘. Let N, L ≤ M. N is said to be a δ-supplement [3, 10] of L in M if N + L = M and N ∩ L δ N. M is called a δ-supplemented module if every submodule of M has a δ-supplement. An epimorphism g : Q → M is

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called a projective δ-cover [14] if Q is a projective module and Kerg is δ-small in Q. All undefined concepts can be found in [1, 4, 11]. The following lemmas will be used later. Lemma 1.1. (See [14, Lemma 1.5]) Let M and N be modules. 1. δ(M) = {L ≤ M | L is a δ-small submodule of M}. 2. If K δ M and f : M → N is a homomorphism, then f (K) δ N. In particular, if K δ M ≤ N, then K δ N. 3. If every proper submodule of M is contained in a maximal submodule of M, then δ(M) is the unique largest δ-small submodule of M. Lemma 1.2. (See [11, Lemma 2.2]) Let M be a module. Suppose K is a direct summand of M and N ≤ K, then N δ K if and only if N δ M. Lemma 1.3. (See [14, Lemma 1.2]) Let N be a submodule of M. The following statements are equivalent: 1. N δ M; 2. If X + N = M, then M = X ⊕ Y for a projective semisimple submodule Y with Y ≤ N. 2. Rings characterized by projectivity classes Wang [9] gave characterizations of semiregular rings and (semi-)perfect rings by introducing the concept of projectivity class of modules. Motivated by this, we will characterize δ-semiregular rings and δ-(semi-)perfect rings in term of projectivity classes of modules in this section. Finally, some characterizations of semisimple rings are given. A class P of R-modules is called a projectivity class [9] if it contains all self-projective modules and for every module M and every projective module P with an epimorphism f : P → M, P ⊕ M ∈ P implies that M is projective. Example 2.1. 1. The class of all quasi-projective modules is a projectivity class. 2. The class of all weakly quasi-projective modules in the sense of Rangaswamy and Vanaja is a projectivity class. 3. The class of all pseudo-projective modules is a projectivity class. 4. The class of all direct-projective modules is a projectivity class. 5. For any perfect ring R, the class of all discrete R-modules is a projectivity class. For a projectivity class P, we introduce the following concept. Definition 2.2. We call an epimorphism f : Q → M a P-projective δ-cover of M if Q ∈ P and Kerf is δ-small in Q. The following result generalizes [11, Proposition 4.2]. Where it is proved when P =the class of all direct-projective R-modules and it will be useful for the rest of the paper.

Rings characterized by projectivity classes

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Lemma 2.3. Let P be a projectivity class and which is closed under taking direct summands. Suppose P is a projective module and there is an epimorphism f : P → M. If P ⊕ M has a P-projective δ-cover, then M has a projective δ-cover. Proof. Let g : Q → P ⊕ M be a P-projective δ-cover of P ⊕ M. We have an φg exact sequence 0 → g −1(M) → Q → P → 0, where φ : P ⊕ M → P is the projection map. Since P is projective, Q ∼ = P ⊕g −1 (M) and Ker(φg) = g −1 (M) is a direct summand of Q. Note that Kerg δ Q, and so Kerg δ g −1 (M) g by Lemma 1.2. Clearly, we have an exact sequence 0 →Kerg → g −1 (M) → M → 0. So it suffices to show that g −1 (M) is projective. Since P is projective with an epimorphism f : P → M, and g : g −1(M) → M is an epimorphism, there is a homomorphism α : P → g −1(M) such that gα = f , and hence Imα+Kerg = g −1 (M). Since Kerg is δ-small in g −1 (M), there is a projective semisimple submodule L of Kerg such that Imα ⊕ L = g −1 (M) by Lemma 1.3. Thus Q ∼ = P ⊕Imα ⊕ L belongs to P. Since P is closed under taking direct summands, P ⊕Imα ∈ P, and there is an epimorphism P →Imα, Imα is projective. Thus g −1 (M) =Imα ⊕ L is projective, as desired. Corollary 2.4. Let P be a projectivity class and which is closed under taking direct summands. Then a projective module P is a δ-semiregular module if and only if P ⊕ (P/N) has a P-projective δ-cover for every finitely generated (cyclic) submodule N of P . Proof. It follows from [14, Lemma 2.4] and Lemma 2.3. Theorem 2.5. Let P be a projectivity class and which is closed under taking direct summands. The following statements are equivalent for a ring R: 1. R is a δ-semiregular ring; 2. Every finitely presented module has a P-projective δ-cover; 3. The module R ⊕ (R/Rr) has a P-projective δ-cover for any r ∈ R. Proof. It is clear by Corollary 2.4 and [11, Theorem 2.10]. Corollary 2.6. ([11, Proposition 4.4]) The following statements are equivalent for a ring R: 1. R is a δ-semiregular ring; 2. Every finitely presented module has a direct-projective δ-cover; 3. The module R ⊕ (R/Rr) has a direct-projective δ-cover for any r ∈ R. A ring R is called δ-(semi)perfect [14] if every (simple) R-module has a projective δ-cover. Recall that a projectivity class P of R-modules satisfies the condition (ME) [9] if the following holds: If F : R − mod → S − mod define a morita equivalence and let S P = {F (A)|A ∈ P}, then S P satisfies the following condition: for every S-module M and every projective S-module P with an epimorphism f : P → M, P ⊕ M ∈S P implies that M is a projective S-module.

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Theorem 2.7. Let P be any projectivity class and which is closed under taking direct summands. The following statements are equivalent for a ring R: 1. R is a left δ-semiperfect ring; 2. Every finitely generated left R-module has a P-projective δ-cover; 3. Every 2-generated left R-module has a P-projective δ-cover; If P satisfies the condition (ME), (1) − (3) are equivalent to: 4. For all natural number n, every cyclic left Rn -module has a P-projective δ-cover, where Rn denotes the ring of all n by n matrices over R; 5. There exists a natural number n > 1 such that every cyclic left Rn -module has a P-projective δ-cover. Proof. The implications (1) ⇒ (2) ⇒ (3) and (1) ⇒ (4) ⇒ (5) are trivial. (3) ⇒ (1) Let I be a left ideal of R. Then (R ⊕ R/I) has a P-projective δ-cover by hypothesis. By Lemma 2.3, every cyclic R-module has a projective δ-cover, and hence R is a left δ-semiperfect ring. (5) ⇒ (1) Let I be a left ideal of R. It suffices to show that R/I has a P-projective δ-cover. We denote by In the left ideal of Rn consisting of all matrices with entries from I, then In is a left ideal of Rn . Let eij ∈ Rn be the matrix unitis. ThenRn /In eij ∼ = P ⊕ M as left Rn -modules, where M = Rn e11 /In e11 and P = ni=2 Rn eii . Clear, P is projective and there is an Rn -epimorphism P → M via P (rij )ni,j=1 → (rij )ni,j=1e21 +In e11 . Hence cyclic Rn -module Rn /In e11 (∼ = P ⊕ M) has a P-projective δ-cover. By Lemma 2.3, Rn -module M has a projective δ-cover f : Q → M. Since e11 Rn e11 ∼ = R and R/I as R-modules. Since Q is a projective Rn f (e11 Q) = e11 f (Q) = e11 M ∼ = module, e11 Q is a projective R-module, and hence the R-module epimorphism e11 Q → R/I induced by f is a projective δ-cover of cyclic R-module R/I. Thus R is a left δ-semiperfect ring. Corollary 2.8. ([14, Theorem 3.6]) A ring R is a δ-semiperfect ring if and only if every finitely generated module has a projective δ-cover. Theorem 2.9. Let P be any projectivity class and which is closed under taking direct summands. Then a ring R is δ-perfect if and only if every R-module has a P-projective δ-cover. Proof. “ ⇒ ” is clear. “ ⇐ ” Let M be any R-module. Then there is a free module F with an epimorphism f : F → M. Since every R-module has a P-projective δ-cover, F ⊕ M has a P-projective δ-cover. Since F is projective, M has a projective δ-cover by Lemma 2.3, and hence R is δ-perfect. Corollary 2.10. A ring R is δ-perfect if and only if every R-module has a direct-projective δ-cover. Recall that a class I of R-modules is called an injectivity class [8] if it is closed under direct summands, contains all self-injective (= quasi-injective)


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modules and M ⊕ E(M) ∈ I implies that M is injective. It is well known that the class of all quasi-injective modules, the class of all (quasi-)continuous modules and the class of all direct-injective modules are all injectivity classes. Let R be a ring and C a class of R-modules, C is said to be socle fine [2] whenever for any M, N ∈ C we have Soc(M) Soc(N ) if and only if M N. Theorem 2.11. Let R be a ring, I any injectivity class and P any projectivity class. Then the following statements are equivalent: 1. R is semisimple; 2. P is socle fine; 3. I is socle fine. Proof. If R is semisimple, then the class of all R-modules is socle fine and hence (1) implies (2), (3). (2) ⇒ (1) Since R and Soc(R) belong to P and Soc(R) = Soc(Soc(R)), we have R Soc(R) by (2). Thus R is semisimple. (3) ⇒ (1) Since Soc(E(R)) = Soc(R) = Soc(Soc(R)) and both of E(R) and Soc(R) are in I, we have E(R) Soc(R) is semisimple, and so is R. Corollary 2.12. ([2, Theorem 2.1]) The following assertions are equivalent for a ring R: 1. R is semisimple; 2. The class of all quasi-projective modules is socle fine; 3. The class of all quasi-injective modules is socle fine. Lemma 2.13. ([13, Theorem 9]) The following statements are equivalent for a ring R: 1. R is semisimple; 2. The direct sum of two direct-projective R-modules is direct-projective; 3. The R-module RR is direct-injective and the direct sum of two directinjective R-modules is direct-injective. Corollary 2.14. The following statements are equivalent for a ring R: 1. R is semisimple; 2. The class of all direct-projective modules is socle fine. 3. The class of all direct-injective modules is socle fine. 4. The class C of all finite direct sums of direct-projective modules is socle fine. If the R-module RR is direct-injective, then (1)−(4) are equivalent to (5). 5. The class C of all finite direct sums of direct-injective modules is socle fine. Proof. (1) ⇔ (2) ⇔ (3) are clear by Theorem 2.11. (1) ⇒ (4) and (1) ⇒ (5) are straightforward. (4) ⇒ (1) We shall prove that R is semisimple by showing that the class of the direct-projective R-modules is closed under finite direct sums. Let P1 , P2 ·

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··, Pn be direct-projective modules and P = ⊕i Pi be an element in C, let Q = Soc(P ) which is direct-projective by its semisimple character. Since P, Q ∈ C and Soc(P ) = Soc(Q), we infer that P Q and hence P is semisimple. Thus P is direct-projective. Now the fact that the class of direct-projective modules is closed under finite direct sum implies R is semisimple by Lemma 2.13. (5) ⇒ (1) is similar to (4) ⇒ (1).

3. A characterization of δ-semiregular rings Recall that a module M is called a finitely δ-supplemented module [11] if every finitely generated submodule of M has a δ-supplement in M, i.e., for every finitely generated submodule N of M, there exists L ≤ M such that M = N + L and N ∩ L δ L. M is said to be amply finitely δ-supplemented [11] if M is finitely δ-supplemented, and for any pair of submodules U and V of M such that U +V = M and U is finitely generated, there is a δ-supplement V of U in M with V ≤ V . M is called a δ-semiregular module [11] if for every finitely generated submodule N of M, there is a decomposition M = K ⊕ K such that K ≤ N and N ∩ K δ K . It is well known that [7] a ring R is a semiregular ring if and only if RR is an amply finitely supplemented module if and only if RR is a finitely supplemented module. We prove that a ring R is a δ-semiregular ring if and only if RR is an amply finitely δ-supplemented module if and only if RR is a finitely δsupplemented module in this section. Theorem 3.1. The following statements are equivalent for a finitely generated projective module P : 1. P is a δ-semiregular module; 2. P is an amply finitely δ-supplemented module; 3. P is a finitely δ-supplemented module. Proof. “(1) ⇒ (2)” Let M be a δ-semiregular module. Suppose that U is a finitely generated submodule of M and V a submodule of M with U + V = M. Since M is a δ-semiregular module, there is a decomposition M = U ⊕ Q such that U ≤ U and U ∩ Q δ Q. Then U = U + (U ∩ Q) and U ∩ Q δ Q. Therefore M = V + U + (U ∩ Q). Since U ∩ Q δ M, there is a projective semisimple submodule P of U ∩ Q such that M = (V + U ) ⊕ P by Lemma 1.3. Since M is projective, V + U is projective. Therefore there is V ≤ V such that V + U = V ⊕ U , and hence M = V ⊕ U ⊕ P . Let f : M → V be the projection. Since U ⊕ P ≤ U and M = U ⊕ Q = U ⊕ P ⊕ V , U = U ⊕ (U ∩ Q) = (U ⊕ P ) ⊕ (U ∩ V ). Therefore f (U ∩ Q) = f (U) = U ∩ V . Since U ∩ Q δ M, U ∩ V = f (U ∩ Q) is δ-small in V by Lemma 1.1. Note that M = U + V (for U ⊕ P ≤ U), and so V is a δ-supplement of U in M with V ≤ V .. “(2) ⇒ (3)” is clear.


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“(3) ⇒ (1)” It suffices to prove that for every finitely generated submodule N of P , P/N has a projective δ-cover by [14, Lemma 2.4]. Let N be a finitely generated submodule of P . Since P is a finitely δ-supplemented module, there exists a submodule X of P such that P = X + N and X N δ X. Let π : P → P/N be the canonical epimorphism and π |X : X → P/N be the restriction of π to X. Since P is projective, there is a morphism g : P → X such Since that π = π |X ·g. Then X = g(P ) + (X N), and hence P = g(P ) + N. X N δ X, there exists a projective semisimple submodule Y ≤ (X N) such that X = g(P ) Y by Lemma 1.3. We will show that (g(P ) N) δ g(P ). Suppose that (g(P ) N) + Z = g(P ) with g(P )/Z singular. Then X = (g(P ) N) + Z) Y , and Y ) ((g(P ) N) + Z)/Z is so X/(Z Y = g(P ) Y . Thus singular. Since g(P ) N ≤X N δ X, X = Z Z = g(P ), and hence g(P ) N δ g(P ). Therefore g(P ) is a δ-supplement of N in P . Next, we show that g(P ) is projective. Since P is a finitely δsupplemented module, g(P ) has a δ-supplement Q in P . So g(P ) is also a δ-supplement of Q in P by [3, Lemma 2.11]. Then g(P ) is projective by [3, Lemma 2.15]. Since g(P ) N δ g(P ), g(P ) is a projective δ-cover of P/N. As desired. A ring R is said to be δ-semiregular (see [14]) if R/δ(R) is a regular ring and idempotents lift modulo δ(R). Corollary 3.2. The following statements are equivalent for a ring R: 1. 2. 3. 4. 5.

R is a δ-semiregular ring; RR is an amply finitely δ-supplemented module; R R is an amply finitely δ-supplemented module; RR is a finitely δ-supplemented module. R R is a finitely δ-supplemented module.

Proof. It follows from Theorem 3.1 and [11, Theorem 2.10]. The following example shows that a finitely δ-supplemented module need not be a δ-supplemented module. Fi = Z Example 3.3. Let Q = Π∞ 2 . Let R be the subring i=1 Fi , where each a b | a ∈ R, b ∈ Soc(R)}. of Q generated by ⊕∞ i=1 Fi and 1Q and T = { 0 a Then T is a ring under the matrix addition and multiplication. Following by [14, Example 4.2], T is not a δ-semiperfect, and hence T T is not a δsupplemented module by [3, Theorem 3.3]. However, T is a δ-semiregular ring by [14, Example 4.2]. So T T is a finitely δ-supplemented module by Corollary 3.2.

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Received: January, 2010