1 Introduction and Preliminaries

Throughout this paper, is an associative ring with identity and Mod- is the category of unitary right -modules. For a right -module , let ൌ ሺሻ be the endomorphism ring of . A right -module is called -generated if there exists an epimorphism ሺூሻ → for some index set . If is finite, then is called finitely -generated. In particular, is called cyclic if it is isomorphic to / for some submodule of or equivalent to saying that any -cyclic submodule of can be considered the image of an endomorphism of . Following Wisbauer [1], ሾሿ denotes the full subcategory of Mod- whose objects are submodules of -generated modules. A right -module is called a self-generator if it generates all of its submodules. A right -module is called a subgenerator if it is a generator of ሾሿ. For undefined notation, terminology and all the basic results on rings and modules see [1-3].

In 1976, Zelmanowitz [4] introduced the notion of compressible modules. A right -module is called compressible if for each nonzero submodule of there exists an -module monomorphism from to . For example, if is a domain, then the right -module is compressible. Generalizations of compressible modules have been studied in several papers (see [5-7]). Recently, P.F. Smith [8] introduced the concept of a slightly compressible module, which is a generalization of the compressible module. According to P.F. Smith, a right -module is called slightly compressible if for any nonzero submodule of there exists a nonzero -module homomorphism from to . See for example [8], Example 1.2: if is a nonzero ring and is the ring of 22 upper triangular matrices over , then the right -module is slightly compressible.

In this paper, the notion of -slightly compressible modules where is a right module is introduced and studied, which is a general form of slightly compressible modules. Moreover we provide conditions for any right -module to be an slightly compressible module and an example of -slightly compressible modules. Some results on slightly compressible modules [8] are extended to -slightly compressible modules.

2 -slightly Compressible Modules

In this section, we introduce the concept of -slightly compressible modules. We investigate the basic properties of -slightly compressible modules. Some of these properties are analogous to the properties of slightly compressible modules. First, we give the following definition:

Definition 2.1 Let and be right -modules. is called -slightly compressible if for every nonzero submodule of there exists a nonzero -homomorphism from to such that ሺሻ ↪ . In the case that ൌ, is called a slightly compressible module, referring to [8].

Example 2.2

• (1) This example is taken from [9]. A right -module is called fully- cyclic if for every submodule of there exists ∈ ோሺ, ሻ such that ൌ ሺሻ. A right -module is called quasi-fully-cyclic if it is a fully--cyclic module. It is clear that every fully--cyclic module is an -slightly compressible module.
• (2) Let and be right -modules. If is an -generated module, then is an slightly compressible module (see, [[3], Exercise 2(b) and (d), pp. 361-362]).

(3) Let F be a field and \(R = \begin{pmatrix} F & F \\ 0 & F \end{pmatrix}\) the ring of all matrices of the form \(\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}\) where \(a, b, c \in F\), \(M_R = \begin{pmatrix} F & F \\ 0 & F \end{pmatrix}\) and \(N_R = \begin{pmatrix} 0 & F \\ 0 & 0 \end{pmatrix}\). Then, \(M_R\) and \(N_R\) are \(R_R\)-slightly compressible modules.

Theorem 2.3 Let M be a Noetherian right R-module. If N is an M-slightly compressible module, then \(Soc(M) \cong Soc(N)\).

Proof. Assume that N is an M-slightly compressible module. Let A be a simple submodule of N. There exists \(0 \neq s \in Hom_R(M, N)\) such that \(0 \neq s(M) \hookrightarrow A\). But A is a simple submodule of N, A = s(M). Let \(0 \neq a \in A\). Then, aR = A = s(M), so a = s(b) for some \(b \in M\). In Noetherian module bR, there exists a simple submodule B containing b such that \(A \cong B\). Therefore \(Soc(M) \cong Soc(N)\).

Theorem 2.4 Let M, M' and N be right R-modules, where N is an M-slightly compressible module.

• (1) If M is an epimorphism image of M', then N is an M'-slightly compressible module.
• (2) If M is an M'-slightly compressible module, then N is also an M'-slightly compressible module.
• (3) For any submodule A of N, A is an essential in N if and only if for any \(0 \neq t \in Hom_{\mathbb{R}}(M, N), t(M) \cap A \neq 0\).
• (4) For any submodule A of N, A is an uniform submodule of N if and only if for any \(0 \neq t \in Hom_R(M, A)\), t(M) is an essential in A.

Proof.

• (1) Assume that M is an epimorphism image of M'. There exists an epimorphism \(\alpha\) from M' to M, so \(\alpha(M') = M\). Let \(0 \neq A \hookrightarrow N\). Since N is an M-slightly compressible, there exists \(0 \neq s \in Hom_R(M,N)\) such that \(s(M) \hookrightarrow A\). Thus \(s\alpha(M') \hookrightarrow A\). Therefore N is an M'-slightly compressible module.
• (2) Assume that M is an M'-slightly compressible module. Let \(0 \neq A \hookrightarrow N\). Since N is an M-slightly compressible module, there exists \(0 \neq s \in Hom_R(M,N)\) such that \(s(M) \hookrightarrow A\). Because M is an M'-slightly compressible module, there exists \(0 \neq t \in Hom_R(M',M)\) such that \(t(M') \hookrightarrow M\). Then, \(st(M') \hookrightarrow s(M) \hookrightarrow A\). Thus N is an M'-slightly compressible module.
• (3) \((\Rightarrow)\) It is obvious.

• ሺ⇐ሻ Assume that any 0 ് ∈ ோሺ, ሻ, ሺሻ ∩ ് 0 holds. Let 0്↪. Since is an -slightly compressible module, there exists 0 ് ∈ ோሺ, ሻ such that ሺሻ ↪ . Thus ሺሻ ∩്0 and we have ∩്0. Therefore is an essential in .
• ሺ4ሻ ሺ⇒ሻ It is clear.
  • ሺ⇐ሻ Assume that for any 0 ് ∈ ோሺ, ሻ such that ሺሻ is an essential in . Let and be nonzero submodules of . Since is an slightly compressible module, there exists , ∈ ோሺ, ሻ such that 0്ሺሻ ↪ and 0 ് ሺሻ ↪ . By assumption we have ሺሻ and ሺሻ are essential in . Then, ሺሻ ∩ ሺሻ ് 0 and we have ∩്0. Therefore is uniform.

Proposition 2.5 Let and be right -modules such that ோሺ, ሻ ് 0. Then, is a simple module if and only if is an -slightly compressible module with every nonzero R-homomorphism from to is an epimorphism.

Proof.

• ሺ⇒ሻ It is obvious.
• ሺ⇐ሻ Assume that is an -slightly compressible module with every nonzero R-homomorphism from to is an epimorphism. Let 0 ് ↪ . There exists 0 ് ∈ ൌ ோሺ, ሻ such that 0്ሺሻ ↪ . By assumption we have ൌ ሺሻ and hence ൌ. Therefore is a simple module.

Corollary 2.6 ([10], Proposition 3.5) Let be a right -module. Then, is a simple module if and only if is a slightly compressible module with every nonzero endomorphism of is an epimorphism.

Proposition 2.7 Let be an -slightly compressible module. Then,

• (1) is an -slightly compressible module for all ↪ .
• (2) is an -slightly compressible module for every right -module with ሺሻ ് ↪ for all ∈ ோሺ, ሻ.

Proof.

• (1) Let ↪. If ൌ0, it is clear. We can suppose that ്0. Let 0്↪ , Then, ↪ and there exists 0 ് ோሺ, ሻ such that ሺሻ ↪ . Thus, 0 ് ∈ ோሺ, ሻ. Hence, is an -slightly compressible.
• (2) Let ↪ such that ሺሻ ് for all ∈ ோሺ, ሻ. Let ്0↪. Since is an -slightly compressible module, there exists 0്∈

ோሺ, ሻ such that ሺሻ ↪ and ሺሻ ് . Then, 0 ് | ∈ ோሺ, ሻ such that, |ሺሻ ↪ , where | is an -homomorphism with respect to . Therefore is an -slightly compressible.

Proposition 2.8 Let and be right -modules. If every nonzero submodule of containing nonzero submodule such that ≅ where is a direct summand of , then is an -slightly compressible module.

Proof. Assume that every nonzero submodule of containing nonzero submodule such that ≅ where is a direct summand of . Let 0്↪ . By assumption there exists a nonzero submodule such that ≅ where is a direct summand of . Since ≅ there exists a that is an isomorphism from to . Let be the canonical projection map from to . Thus, : is an -homomorphism and ሺሻ ↪ . Therefore is an slightly compressible module.

Recall that ∈ሾሿ is called hereditary in ሾሿ if every submodule of is a projective in ሾሿ. We say that a ring is right (left) hereditary if ோሺோ) is a hereditary in Mod-.

Theorem 2.9 Let be a right hereditary ring and an injective right -module. If is an -slightly compressible module, then every nonzero submodule of contains a direct summand of .

Proof. Assume that is an -slightly compressible module. Let 0്↪. By assumption there exists 0 ് ∈ ோሺ, ሻ such that ሺሻ ↪ . Since is an injective module, is a hereditary ring and Theorem 3.22 in [11], |୩ୣ୰ ሺ௦ሻ is an injective. But |୩ୣ୰ ሺ௦ሻ ≅ ሺሻ, ሺሻ is an injective. Therefore ሺሻ is a direct summand of .

Proposition 2.10 Let and be right -modules such that is an -slightly compressible module. If every -cyclic submodule of is an injective module, then is an -generated module.

Proof. Assume that every -cyclic submodule of is an injective. Let 0്↪. There exists 0 ് ∈ ோሺ, ሻ such that 0 ് ሺሻ ↪ . By assumption ሺሻ is an injective and we have ሺሻ is a direct summand of . There exists ↪ such that ሺሻ ⊕ൌ. If ൌ0, we are done. If 0 ് ↪ , there exists 0 ് ∈ ோሺ, ሻ such that 0്ሺሻ ↪ . By assumption, ሺሻ is an injective and we have ሺሻ is a direct summand of . Thus, there exists ↪ such that ሺሻ ⊕ൌ. Continuous in this process we have ൌ ∑௦∈ு ሺሻ ೃሺெ,ேሻ . Therefore is an -generated module.

Corollary 2.11 Let be a slightly compressible module. If every -cyclic submodule of is an injective then is a self-generator.

Acknowledgements

The authors would like to thank the referee for his/her useful suggestions that improved the presentation of this article. The second author was supported by the Center of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand.