On strongly coseparable modules

A module M is called s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that V⊆U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V \subseteq U$$\end{document} and M/V is finitely generated. It is shown that every non-finitely generated free module is s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable but a finitely generated free module is not, in general, s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable. We prove that the class of s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {s}$$\end{document}-coseparable are provided.

subclass of that of coseparable modules. But we prove that for a non-finitely generated module M, M is coseparable if and only if M is s-coseparable. It is shown that a direct summand of an s-coseparable module is not, in general, s-coseparable. On the other hand, we show that the class of finitely generated s-coseparable modules is closed under direct summands. Moreover, we prove that the class of s-coseparable modules over a right noetherian ring is closed under finite direct sums. Section 3 is devoted to shed some light on a proper subclass of that of s-coseparable modules. A module M is called completely s-coseparable if every direct summand of M is s-coseparable. Some examples of completely s-coseparable modules are provided.
In Sect. 4, we characterize several classes of rings in terms of s-coseparable modules. The class of rings R for which every finitely generated projective R-module is s-coseparable, is shown to be exactly that of the weakly regular semiprimitive rings; while the class of commutative rings R for which every cyclic R-module is s-coseparable is characterized as that of the von Neumann regular rings.

Some examples and properties of strongly coseparable modules
Recall that a module M is called coseparable if for every submodule U of M such that M/U is finitely generated, there exists a direct summand V of M such that V ⊆ U and M/V is finitely generated (see e.g., [9,12,19]). It is clear that a finitely generated module is always coseparable. In this paper, we introduce and study a subclass of coseparable modules as follows: Definition 2.1 A module M is said to be strongly coseparable (or s-coseparable for short) if for every nonzero submodule U of M with the property that M/U is finitely generated, there exists a nonzero direct summand U of M such that U ⊆ U and M/U is finitely generated.
A submodule of a module M is called cofinite if M/N is finitely generated. It is easily seen that any semisimple module is s-coseparable. In addition, any module M with Rad(M) = M is s-coseparable. In this section, we present more examples of s-coseparable modules. Recall that a module M is called regular if every finitely generated submodule of M is a direct summand.

Example 2.3
Let M = x 1 R + · · · + x n R be a finitely generated regular module. Let U be a nonzero cofinite submodule of M. Since Rad(M) = 0, there exists a proper submodule N of M such that M = U + N . For each i ∈ {1, . . . , n}, there exist u i ∈ U and y i ∈ N such that x i = u i + y i . Let U be the submodule of U generated by {u 1 , . . . , u n }. Since M = U + N and N = M, we have U = 0. Moreover, U is a direct summand of M as M is regular. Clearly, M/U is finitely generated. Therefore, M is s-coseparable.
In particular, the right R-module R R is s-coseparable for every von Neumann regular ring R.
A module M is said to be A-projective if for every submodule X of A, any homomorphism ϕ : M → A/ X can be lifted to a homomorphism ψ : M → A. Example 2.4 (i) Let a module M = M 1 ⊕ M 2 be a direct sum of a finitely generated submodule M 1 and a semisimple submodule M 2 which is not finitely generated. Assume that M 2 is M 1 -projective (e.g., M 2 can be projective). We want to prove that M is s-coseparable. To do this, let U be a cofinite submodule of M. Then M = U +K for some finitely generated submodule K of M. Note that K ⊆ M 1 ⊕K 2 for some finitely generated direct summand K 2 of M 2 . Hence, M = U + (M 1 ⊕ K 2 ). Moreover, we have M = L 2 ⊕ (M 1 ⊕ K 2 ) for some submodule L 2 of M 2 . Using [6, 4.3], it follows that L 2 is (M 1 ⊕K 2 )-projective. Therefore, M = U ⊕(M 1 ⊕K 2 ) for some submodule U of U by [6, 4.12]. Clearly, U = 0 as M is not finitely generated. In addition, M/U is finitely generated. Consequently, M is an s-coseparable module.
(ii) Let T be a commutative ring and let m be a maximal ideal of T . Consider the ring R = T × T /m. Taking (i) into account, it follows that for any finitely generated R-module M 1 In the following proposition, we determine the structure of the indecomposable s-coseparable modules. It is easily seen that every s-coseparable module is coseparable. But the converse is not true, in general, as illustrated in the next example.
Example 2.6 Let M be an indecomposable finitely generated module which is not simple (e.g., we can take M = R where R is any commutative integral domain which is not a field or M = Z/ p n Z for some prime number p and an integer n ≥ 2). Then M is not s-coseparable by Proposition 2.5. On the other hand, M is coseparable since M is finitely generated.
A module M is called weakly regular (or an I 0 -module) if every submodule, which is not contained in Rad(M), contains a nonzero direct summand of M (see, e.g., [1,16]). A ring R is called weakly regular (or an I 0 -ring as in [16, p. 131]) if the right R-module R R is weakly regular. Equivalently, if the left R-module R R is weakly regular. Many examples of weakly regular rings are given in [16].
The following characterization provides a rich source of examples of s-coseparable modules (see [8]).

Theorem 2.7
The following conditions are equivalent for an R-module M: has the finite exchange property. This is equivalent to saying that for any r ∈ R, there is an idempotent e in R with e ∈ r R and 1 − e ∈ (1 − r )R (see [6, 11.16]).
As an application of the previous proposition, we obtain the following example. Recall that a module M is said to be π-projective if for every two submodules U, It is well known that every quasiprojective module is π-projective (see [18, p. 359]).

Proposition 2.10 Let M be a π-projective module such that every proper finitely generated submodule is contained in a proper finitely generated direct summand of M. Assume that Rad(M) = 0 or M is not finitely generated. Then M is an s-coseparable module.
Proof Case 1: Assume that Rad(M) = 0. Let U be a nonzero cofinite submodule of M. By similar arguments as in the proof of Proposition 2.8, we have M = U + K such that K is a finitely generated proper submodule of M. By hypothesis, there exists a finitely generated proper direct summand L of M such that K ⊆ L. Hence,  (ii) Let M be a π-projective regular module such that Rad(M) = 0 or M is not finitely generated. Then M is s-coseparable by Proposition 2.10.
(iii) Let R be a von Neumann regular ring and let P be a projective R-module. By [16, Proposition 6.7(4)], P is a regular module. Moreover, we have Rad(P) = 0 by [3,Proposition 17.10]. Therefore, P is s-coseparable by (ii).
Next, we exhibit some examples to illustrate that a direct summand of an s-coseparable module is not, in general, s-coseparable.

Example 2.12
Let R be a commutative domain which is not a field.
(i) Assume that Rad(R) = 0 (e.g., we can take R = Z). By the preceding Example, M = R (N) is an s-coseparable R-module. However, the right R-module R R is not s-coseparable by Proposition 2.5.
(ii) Let m be a maximal ideal of R. Consider the ring On the other hand, the T -module R ⊕ 0 is not s-coseparable by Proposition 2.5.
Proof If M is finitely generated, then the result follows from Proposition 2.16(i). Now assume that M is not finitely generated. Let N be a cofinite submodule of M. Then clearly N is not finitely generated. Note that N is coseparable by [8,Proposition 2.15]. Therefore, N is s-coseparable by Theorem 2.7.

Proposition 2.18
Let M be an s-coseparable module. Then the following hold: (ii) Assume that M is fgs-coseparable. Let U be an indecomposable submodule of M. Then U is a direct summand of M by (i). Therefore, U is an fgs-coseparable module by Proposition 2.16(ii). Therefore, M is a simple module by Proposition 2.5.
Next, we provide some sufficient conditions for an fgs-coseparable module to be semisimple. The following corollary is a direct consequence of Proposition 2.18.

Corollary 2.19 Let M be a finitely generated module which is a sum of indecomposable submodules. Then M is fgs-coseparable if and only if M is semisimple.
A module M is said to be nitely cogenerated (or nitely embedded) if there exist nitely many simple modules  A module M is called semiartinian if every nonzero factor module of M has a nonzero socle. A ring R is called a right semiartinian ring if the right R-module R R is semiartinian. The following corollary follows directly from Proposition 2.21.

Corollary 2.22 Let M be a finitely generated R-module which is a semiartinian module with Rad(M) = 0.
Then M is fgs-coseparable.
Recall that a ring R is called a right V-ring if every simple right R-module is an injective module.

Example 2.23
Let R be a right semiartinian right V-ring. Using Proposition 2.21, it follows that every finitely generated R-module is fgs-coseparable.
Next, we focus on the question of when is the direct sum of two s-coseparable modules, s-coseparable? Combining Theorem 2.7 and [1, Theorem 2.9], we obtain the following proposition. Since (M 2 + U )/U is semisimple, it follows that (M 1 + U )/U is a direct summand of M/U . Hence, M 1 /(U ∩ M 1 ) is finitely generated. Case 1: Assume that U ∩ M 1 = 0. Since M 1 is s-coseparable, there exists a nonzero direct summand U 1 of M 1 such that U 1 ⊆ U ∩ M 1 ⊆ U and M 1 /U 1 is finitely generated. This clearly implies that U 1 is a direct summand of M and M/U 1 is finitely generated. Case 2: Assume that U ∩ M 1 = 0. Then M 1 is finitely generated. Using Proposition 2.24, it follows that U contains a nonzero cofinite direct summand of M.

Corollary 2.26 Let R be a right noetherian ring. Then any finite direct sum of s-coseparable R-modules is s-coseparable.
Proof Let M = M 1 ⊕ M 2 be a direct sum of two s-coseparable submodules.

Proposition 2.27 Let M = ⊕ i∈I M i be a direct sum of nonzero submodules M i (i ∈ I ) such that for every submodule N of M, we have N = ⊕ i∈I (N ∩ M i ) (e.g., M is a duo module). Assume that M i is s-coseparable for all i ∈ I . Then M is s-coseparable.
Proof Case 1: Assume that M is finitely generated. Then I is a finite set and each M i (i ∈ I ) is finitely generated. Using Proposition 2.24, we see that M is s-coseparable. Case 2: Assume that M is not finitely generated. Note that M is coseparable by [8,Proposition 3.16]. Hence, M is s-coseparable by Theorem 2.7.

Completely s-coseparable modules Definition 3.1 A module M is called completely s-coseparable if every direct summand of M is s-coseparable.
It is clear that semisimple modules are completely s-coseparable. In addition, every module M with Rad(M) = M is completely s-coseparable.
(ii) By Example 2.12, the Z-module M = Z (N) is s-coseparable. On the other hand, M is not completely s-coseparable since Z is not s-coseparable by Proposition 2.5.
In the next proposition, we provide more examples of completely s-coseparable modules.

Recall that a module M is said to have finite uniform dimension if M does not contain an infinite independent
Next, we present a special subclass of completely s-coseparable modules. Following [4], a module M is called an md-module if every maximal submodule of M is a direct summand of M. This is equivalent to say that every cofinite submodule of M is a direct summand of M by [2,Lemma 2.7]. Therefore, every md-module is s-coseparable. Moreover, the class of md-modules is closed under direct summands by [4,Proposition 2.1]. It follows that every md-module is completely s-coseparable.

Rings whose modules are s-coseparable
In this section, we characterize some classes of rings via s-coseparable modules. A ring R is called semiprimitive if Rad(R) = 0.
We begin with the following result which provides a description of all rings R for which the right R-module R R is s-coseparable.

Theorem 4.1 The following statements are equivalent for a ring R:
(i) The R-module R R is s-coseparable; (ii) R is a weakly regular semiprimitive ring; (iv) Every (finitely generated) free R-module F is s-coseparable; (v) Every finitely generated projective R-module is s-coseparable; (vi) For every finitely generated projective R-module P, End R (P) is a weakly regular semiprimitive ring.
Proof (i) ⇒ (iv) Let F be a free R-module. Then, Rad(F) = 0 by Theorem 2.7. If F is not finitely generated, then F is s-coseparable by Example 2.11(i). Now if F is finitely generated, then F is s-coseparable by Proposition 2.24.
(iv) ⇒ (v) Let P be a finitely generated projective R-module. It is well known that P is isomorphic to a direct summand of a finitely generated free R-module F. Therefore, P is s-coseparable by Proposition 2.16.
(vi) ⇒ (ii) This follows from the fact that the rings End R (R R ) and R are isomorphic.
The next corollary follows directly from Theorem 4.1.

Corollary 4.2 For any ring R, the right R-module R R is s-coseparable if and only if it is completely scoseparable.
Next, we shed some light on the class of rings R for which every cyclic R-module is s-coseparable.

Proposition 4.3 Let R be a ring such that every cyclic R-module is s-coseparable. Then, R is a right V-ring.
Proof Let I be a right ideal of R. By hypothesis, R/I is an s-coseparable R-module. Hence, Rad(R/I ) = 0 by Theorem 2.7. Therefore, R is a right V-ring by [13,Theorem 3.75].

Corollary 4.4
The following statements are equivalent for a commutative ring R: (i) Every cyclic R-module is s-coseparable; (ii) R is a von Neumann regular ring.
Proof (i) ⇒ (ii) This follows from Proposition 4.3 and the fact that a commutative ring is von Neumann regular if and only if it is a V-ring (see [13,Corollary 3.73]).
(ii) ⇒ (i) Let I be an ideal of R. Note that R/I is a von Neumann regular ring. Then, every cyclic R-submodule of R/I is a direct summand of the R-module R/I . Therefore, R/I is an s-coseparable R-module.
It is shown in Example 2.12 that the class of s-coseparable modules is not closed under factor modules. In the same vein, the next example shows that a factor module of a cyclic s-coseparable module need not be s-coseparable, in general.
Example 4.5 Let R be a commutative weakly regular semiprimitive ring which is not von Neumann regular. An explicit example of R can be found in [16,Example 15.7(9)]. By Theorem 4.1, the R-module R is scoseparable. On the other hand, using Corollary 4.4, it follows that the ring R has a cyclic R-module M which is not s-coseparable. It is clear that M ∼ = R/I for some nonzero ideal I of R.
Recall that a ring R is called right semiartinian if every nonzero cyclic right R-module has a nonzero socle. The next result provides some information about the class of rings R for which every R-module is scoseparable.

Corollary 4.7
The following statements are equivalent for a ring R: (i) Every R-module is s-coseparable; (ii) R is a weakly regular semiprimitive ring such that every factor module of an s-coseparable module is also s-coseparable.
(ii) ⇒ (i) Let M be an R-module. Then, M ∼ = F/N for some free R-module F and a submodule N of F. By Theorem 4.1, F is s-coseparable. By hypothesis, M is also an s-coseparable module.
Recall that a ring R is said to be right semihereditary if every finitely generated right ideal of R is projective as a right R-module.
From Proposition 2.5 and [3, Proposition 17.14], it follows that every nonzero indecomposable projective module which is not simple could not be s-coseparable. For example, the Z-module Z is not s-coseparable (see also [8,Example 2.11(i)]). Next, we present some examples of classes of rings R for which every projective R-module is s-coseparable.

Proposition 4.8 Let R be a ring. Suppose that one of the following conditions is satisfied:
(i) R is an exchange ring; (ii) R is a right semihereditary ring. Then, every projective R-module is s-coseparable if and only if R is a weakly regular semiprimitive ring.
As an application of Proposition 4.8, we have the following example. Example 4.9 (i) Let T be an exchange ring and consider the ring R = T /Rad(T ). By [16,Theorem 29.2], R is a semiprimitive exchange ring. Moreover, R is weakly regular by [16,Remark 29.7(1)]. Using Proposition 4.8, it follows that every projective R-module is s-coseparable.
(ii) Note that every von Neumann regular ring is a right semihereditary ring which is weakly regular and semiprimitive.
A ring R is called a right max ring if every nonzero right R-module contains a maximal submodule. It is well known that right V-rings and right perfect rings are right max rings (see [3,Theorem 28.4]). Proposition 4.10 Let R be a ring such that Rad(E(S)) = E(S) for each simple R-module S (for instance, R could be a right max ring). Then, the following statements are equivalent: (i) Every finitely cogenerated R-module is s-coseparable; (ii) E(S) is s-coseparable for every simple R-module S; (iii) R is a right V-ring.
(ii) ⇒ (iii) Let S be a simple R-module. Since E(S) is an indecomposable s-coseparable module, it follows that E(S) is simple by Proposition 2.5. Thus S = E(S) is injective. Therefore, R is a right V-ring.
(iii) ⇒ (i) Since R is a right V-ring, every finitely cogenerated R-module is semisimple. The result follows.
A ring R is said to be right small if the right R-module R R is a small submodule in its injective hull E(R R ). Note that every commutative domain is a (right) small ring (see [11,Proposition 2.4 and Corollary 2.5]). We conclude this paper by two results devoted to the class of rings R for which every injective R-module is s-coseparable. Let us first give an example of a ring belonging to this class.

Example 4.11
Let R be a right small ring. From [11, Proposition 2.4 and Corollary 2.5], it follows that Rad(E) = E for every injective R-module E. Hence, every injective R-module is s-coseparable.

Proposition 4.12 Let R be a ring and let M be an R-module. Consider the following statements:
(i) R is a right max ring and every R-injective module is s-coseparable; (ii) Every nonzero R-module contains a nonzero finitely generated injective submodule; (iii) R is a right semiartinian right V-ring. Then, (i) ⇒ (ii) ⇒ (iii) hold.
Proof (i) ⇒ (ii) Let M be a nonzero R-module and let E(M) be the injective hull of M. Since R is a right max ring, E(M) has a maximal submodule L. By hypothesis, E(M) is s-coseparable. Then, there exist submodules N and K of E(M) such that E(M) = N ⊕ K , N ⊆ L and K is a nonzero finitely generated submodule of M. But M is essential in E(M). Therefore, M ∩ K = 0. As K is injective, K is s-coseparable. Hence, M ∩ K contains a nonzero direct summand U of K . It is clear that U is a finitely generated injective submodule of M.

Corollary 4.13
Let R be a right perfect ring. Then, the following statements are equivalent: (i) Every injective R-module is s-coseparable; (ii) R is a semisimple ring.
Proof (i) ⇒ (ii) By [3, Theorem 28.4], R is a right max ring and R/Rad(R) is semisimple. By Proposition 4.12, R is a right V-ring. Hence, Rad(R) = 0. It follows that R is a semisimple ring.
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