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On strongly coseparable modules

Abstract

A module M is called \(\mathfrak {s}\)-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 \(V \subseteq U\) and M/V is finitely generated. It is shown that every non-finitely generated free module is \(\mathfrak {s}\)-coseparable but a finitely generated free module is not, in general, \(\mathfrak {s}\)-coseparable. We prove that the class of \(\mathfrak {s}\)-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 \(\mathfrak {s}\)-coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is \(\mathfrak {s}\)-coseparable are provided.

Introduction

Throughout this paper, all rings are associative with an identity element and R denotes such a ring. All modules considered are unital right R-modules, unless otherwise specified. Let M be an R-module. The injective hull of M will be denoted by E(M). The notation \(N \subseteq M\) (respectively, \(N \le M\)) means that N is a subset (respectively, a submodule) of M. By \({\mathbb {N}}\) and \({\mathbb {Z}}\) we denote the set of natural numbers and the ring of integers, respectively.

A module M is called coseparable if every submodule U of M with M/U finitely generated contains a direct summand V of M such that M/V is finitely generated. This concept was initially appeared in 1968 [9] for abelian groups. Then the concept was generalized to modules in 1978 [12] by Hiremath and was further studied by other authors (see e.g., [8, 19]). In this paper, we focus on a subclass of coseparable modules. A module M is said to be strongly coseparable (or \({\mathfrak {s}}\)-coseparable) if for every nonzero submodule U of M with the property that M/U is finitely generated, there exists a nonzero direct summand \(U^{\prime }\) of M such that \(U^{\prime }\subseteq U\) and \(M/U^{\prime }\) is finitely generated.

In Sect. 2, we study some basic properties of \({\mathfrak {s}}\)-coseparable modules. We begin by giving many examples belonging to this class of modules. In particular, we show that the class of \({\mathfrak {s}}\)-coseparable modules is a proper 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 \({\mathfrak {s}}\)-coseparable. It is shown that a direct summand of an \({\mathfrak {s}}\)-coseparable module is not, in general, \({\mathfrak {s}}\)-coseparable. On the other hand, we show that the class of finitely generated \({\mathfrak {s}}\)-coseparable modules is closed under direct summands. Moreover, we prove that the class of \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable modules. A module M is called completely \({\mathfrak {s}}\)-coseparable if every direct summand of M is \({\mathfrak {s}}\)-coseparable. Some examples of completely \({\mathfrak {s}}\)-coseparable modules are provided.

In Sect. 4, we characterize several classes of rings in terms of \({\mathfrak {s}}\)-coseparable modules. The class of rings R for which every finitely generated projective R-module is \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-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 \(V \subseteq U\) 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 \({\mathfrak {s}}\)-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 \(U^{\prime }\) of M such that \(U^{\prime }\subseteq U\) and \(M/U^{\prime }\) is finitely generated.

A submodule of a module M is called cofinite if M/N is finitely generated.

Remark 2.2

It is easily seen that a finitely generated module is \({\mathfrak {s}}\)-coseparable if and only if every nonzero submodule of M contains a nonzero direct summand. A submodule N of a module M is said to be small in M, written \(N<< M\), provided \( N + X, \ne M\) for any proper submodule X of M.

It is easily seen that any semisimple module is \({\mathfrak {s}}\)-coseparable. In addition, any module M with \(Rad(M)=M\) is \({\mathfrak {s}}\)-coseparable. In this section, we present more examples of \({\mathfrak {s}}\)-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 \(M=x_1R + \cdots +x_nR\) 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 \(i \in \{1, \ldots , n\}\), there exist \(u_i \in U\) and \(y_i \in N\) such that \(x_i=u_i+y_i\). Let \(U^{\prime }\) be the submodule of U generated by \(\{u_1, \ldots , u_n\}\). Since \(M=U^{\prime } + N\) and \(N \ne M\), we have \(U^{\prime } \ne 0\). Moreover, \(U^{\prime }\) is a direct summand of M as M is regular. Clearly, \(M/U^{\prime }\) is finitely generated. Therefore, M is \({\mathfrak {s}}\)-coseparable.

In particular, the right R-module \(R_R\) is \({\mathfrak {s}}\)-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 \(\varphi :M \rightarrow A/X\) can be lifted to a homomorphism \(\psi :M \rightarrow A\).

Example 2.4

(i) Let a module \(M=M_1 \oplus M_2\) 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 \({\mathfrak {s}}\)-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 \(K \subseteq M_1 \oplus K_2\) for some finitely generated direct summand \(K_2\) of \(M_2\). Hence, \(M = U + (M_1 \oplus K_2)\). Moreover, we have \(M=L_2 \oplus (M_1 \oplus K_2)\) for some submodule \(L_2\) of \(M_2\). Using [6, 4.3], it follows that \(L_2\) is \((M_1 \oplus K_2)\)-projective. Therefore, \(M=U^{\prime } \oplus (M_1 \oplus K_2)\) for some submodule \(U^{\prime }\) of U by [6, 4.12]. Clearly, \(U^{\prime } \ne 0\) as M is not finitely generated. In addition, \(M/U^{\prime }\) is finitely generated. Consequently, M is an \({\mathfrak {s}}\)-coseparable module.

(ii) Let T be a commutative ring and let \({\mathfrak {m}}\) be a maximal ideal of T. Consider the ring \(R=T \times T/{\mathfrak {m}}\). Taking (i) into account, it follows that for any finitely generated R-module \(M_1\), the R-module \(M_1 \oplus (0 \oplus T/{\mathfrak {m}})^{({\mathbb {N}})}\) is \({\mathfrak {s}}\)-coseparable.

In the following proposition, we determine the structure of the indecomposable \({\mathfrak {s}}\)-coseparable modules.

Proposition 2.5

Let M be an indecomposable module. Then the following statements are equivalent:

(i) M is an \({\mathfrak {s}}\)-coseparable module;

(ii) \(Rad(M)=M\) or M is a simple module.

Proof

(i)\(\Rightarrow \)(ii) Assume that \(Rad(M) \ne M\). Then M contains a maximal submodule N. Suppose that \(N \ne 0\). Since M is \({\mathfrak {s}}\)-coseparable, there exists a nonzero direct summand \(N^{\prime }\) of M such that \(N^{\prime } \subseteq N\). Hence, \(N^{\prime } = M\) as M is indecomposable. This contradicts the fact that \(N \ne M\). Therefore, \(N=0\) and hence M is a simple module.

(ii)\(\Rightarrow \)(i) This is immediate. \(\square \)

It is easily seen that every \({\mathfrak {s}}\)-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 \(M={\mathbb {Z}}/p^{n}{\mathbb {Z}}\) for some prime number p and an integer \(n \ge 2\)). Then M is not \({\mathfrak {s}}\)-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 \(_RR\) is weakly regular. Many examples of weakly regular rings are given in [16].

The following characterization provides a rich source of examples of \({\mathfrak {s}}\)-coseparable modules (see [8]).

Theorem 2.7

The following conditions are equivalent for an R-module M:

(i):

M is \({\mathfrak {s}}\)-coseparable;

(ii):

Either (a) M is coseparable and not finitely generated, or (b) \(Rad(M)=0\) and M is a finitely generated module which is weakly regular.

Proof

(i) \(\Rightarrow \) (ii) Clearly, M is coseparable. Assume that M is finitely generated. Suppose that \(Rad(M) \ne 0\). By hypothesis, Rad(M) contains a nonzero direct summand U of M. But \(Rad(M) \ll M\). Therefore, \(U \ll M\) and hence \(U=0\), a contradiction. It follows that \(Rad(M)=0\). It is clear that M is weakly regular.

(a) \(\Rightarrow \) (i) Let U be a nonzero cofinite submodule of M. As M is coseparable, U contains a direct summand \(U^{\prime }\) of M such that \(M/U^{\prime }\) is finitely generated. Thus \(U^{\prime } \not = 0\) since M is not finitely generated. It follows that M is \({\mathfrak {s}}\)-coseparable.

(b) \(\Rightarrow \) (i) This is obvious. \(\square \)

Following [6, 11.26], a module M is called refinable if, for any submodules N and L of M with \(N+L=M\), there exists a direct summand \(N^{\prime }\) of M with \(N^{\prime } \subseteq N\) and \(N^{\prime }+L=M\).

Proposition 2.8

Let M be a refinable module. Assume that \(Rad(M)=0\) or M is not finitely generated. Then M is \({\mathfrak {s}}\)-coseparable.

Proof

Case 1: Assume that \(Rad(M)=0\). Let U be a nonzero cofinite submodule of M. Since \(Rad(M)=0\), U is not small in M. Thus \(M=U+N\) for some proper submodule N of M. Note that \(N/(U\cap N) \cong M/U\) is finitely generated. Then \(N=(U\cap N)+ K\) for some finitely generated submodule K of N. Therefore, \(M=U + K\). Note that \(K\ne M\) since otherwise \(N=M\). Since M is a refinable module, there exist a direct summand \(U^{\prime }\) of M such that \(U^{\prime } \subseteq U\) and \(M=U^{\prime }+K\). Notice that \(U^{\prime } \ne 0\) and \(M/U^{\prime }\) is finitely generated. It follows that M is \({\mathfrak {s}}\)-coseparable.

Case 2: Assume that M is not finitely generated. Then M is \({\mathfrak {s}}\)-coseparable by [8, Example 2.3] and Theorem 2.7. \(\square \)

A module M is said to have the finite exchange property if for any finite index set I, whenever \(M \oplus N = \oplus _{i \in I}A_i\) for modules N and \(A_i\), then \(M \oplus N=M \oplus (\oplus _{i \in I}B_i)\) for submodules \(B_i \le A_i\). A ring R is called an exchange ring if the R-module \(R_R\) has the finite exchange property. This is equivalent to saying that for any \(r \in R\), there is an idempotent e in R with \(e \in rR\) and \(1-e \in (1-r)R\) (see [6, 11.16]).

As an application of the previous proposition, we obtain the following example.

Example 2.9

Let M be a quasi-projective module having the finite exchange property with \(Rad(M)=0\). For example, we can take the T-module \(M=R/Rad(R)\) where R is an exchange ring and \(T=R/Rad(R)\). Then M is \({\mathfrak {s}}\)-coseparable by Proposition 2.8 and [6, 11.31]. Using [16, Proposition 28.6], it follows that every finitely generated projective module M with \(Rad(M)=0\) over an exchange ring is \({\mathfrak {s}}\)-coseparable.

Recall that a module M is said to be \(\pi \)-projective if for every two submodules UV of M with \(U+V=M\) there exists \(f \in End_R(M)\) such that \(\text{ Im }(f) \subseteq U\) and \(\text{ Im }(1-f) \subseteq V\). It is well known that every quasi-projective module is \(\pi \)-projective (see [18, p. 359]).

Proposition 2.10

Let M be a \(\pi \)-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 \({\mathfrak {s}}\)-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 \(K \subseteq L\). Hence, \(M=U+L\). Since M is \(\pi \)-projective, there exists a submodule \(U^{\prime }\) contained in U such that \(M=U^{\prime }\oplus L\) by [18, 41.14(3)]. Since \(L\ne M\), we have \(U^{\prime }\ne 0\). Hence, M is \({\mathfrak {s}}\)-coseparable.

Case 2: Assume that M is not finitely generated. Then M is \({\mathfrak {s}}\)-coseparable by [8, Proposition 2.5] and Theorem 2.7. \(\square \)

Example 2.11

(i) Let R be a ring and consider the R-module \(M=R^{(I)}\), where I is an infinite index set. From Proposition 2.10, it follows that M is \({\mathfrak {s}}\)-coseparable.

(ii) Let M be a \(\pi \)-projective regular module such that \(Rad(M)=0\) or M is not finitely generated. Then M is \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable by (ii).

Next, we exhibit some examples to illustrate that a direct summand of an \({\mathfrak {s}}\)-coseparable module is not, in general, \({\mathfrak {s}}\)-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 \(R={\mathbb {Z}}\)). By the preceding Example, \(M=R^{({\mathbb {N}})}\) is an \({\mathfrak {s}}\)-coseparable R-module. However, the right R-module \(R_R\) is not \({\mathfrak {s}}\)-coseparable by Proposition 2.5.

(ii) Let \({\mathfrak {m}}\) be a maximal ideal of R. Consider the ring \(T=R \times R/{\mathfrak {m}}\). By Example 2.4(ii), the T-module \((R \oplus 0) \oplus (0 \oplus R/{\mathfrak {m}})^{({\mathbb {N}})}\) is \({\mathfrak {s}}\)-coseparable. On the other hand, the T-module \(R \oplus 0\) is not \({\mathfrak {s}}\)-coseparable by Proposition 2.5.

The following two results should be contrasted with Example 2.12.

Proposition 2.13

Let M be an \({\mathfrak {s}}\)-coseparable R-module over a right noetherian ring R. Let N be a fully invariant direct summand of M which is not finitely generated. Then N is \({\mathfrak {s}}\)-coseparable.

Proof

Note that M is coseparable. Then N is coseparable by [8, Proposition 2.12]. Since N is not finitely generated, it follows that N is \({\mathfrak {s}}\)-coseparable by Theorem 2.7. \(\square \)

Combining [8, Proposition 2.16] and Theorem 2.7, we obtain the following result.

Proposition 2.14

Let M be an R-module and let N be a fully invariant submodule of M. Assume that M is \({\mathfrak {s}}\)-coseparable and M/N is not finitely generated. Then M/N is \({\mathfrak {s}}\)-coseparable.

Next, motivated by Theorem 2.7, we will be concerned with finitely generated \({\mathfrak {s}}\)-coseparable modules.

Definition 2.15

A module M is called \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable if M is finitely generated and \({\mathfrak {s}}\)-coseparable.

Proposition 2.16

(i) Any finitely generated submodule of an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module M is also an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module.

(ii) Any direct summand of an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module M is also an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module.

Proof

(i) Let N be a finitely generated submodule of an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module M. Let U be a nonzero submodule of N. Since M is \({\mathfrak {s}}\)-coseparable, there exists a nonzero direct summand \(U^{\prime }\) of M such that \(U^{\prime } \subseteq U\). It is clear that \(U^{\prime }\) is a direct summand of N. Thus N is \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable.

(ii) This follows easily from (i). \(\square \)

Corollary 2.17

Let R be a right noetherian ring and let M be an \({\mathfrak {s}}\)-coseparable R-module. Then every cofinite submodule of M is also \({\mathfrak {s}}\)-coseparable.

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 \({\mathfrak {s}}\)-coseparable by Theorem 2.7. \(\square \)

Proposition 2.18

Let M be an \({\mathfrak {s}}\)-coseparable module. Then the following hold:

(i) Every indecomposable cofinite submodule of M is a direct summand of M.

(ii) If M is finitely generated, then every indecomposable submodule of M is a simple direct summand of M.

Proof

(i) Let U be a nonzero indecomposable cofinite submodule of M. Since M is \({\mathfrak {s}}\)-coseparable, there exists a nonzero direct summand \(U^{\prime }\) of M such that \(U^{\prime } \subseteq U\). By modularity, \(U^{\prime }\) is a direct summand of U. But U is indecomposable. Then \(U=U^{\prime }\) and hence U is a direct summand of M.

(ii) Assume that M is \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable. Let U be an indecomposable submodule of M. Then U is a direct summand of M by (i). Therefore, U is an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module by Proposition 2.16(ii). Therefore, M is a simple module by Proposition 2.5. \(\square \)

Next, we provide some sufficient conditions for an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-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 \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-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 \(S_1, S_2, \ldots , S_k\, \mathrm{such}\ \mathrm{that}\ E(M)\cong E(S_1) \oplus E(S_2) \oplus ... \oplus E(S_k).\)

Corollary 2.20

Let M be a finitely generated R-module. Assume that one of the following conditions is fulfilled:

(i) M has either the ascending chain or the descending chain condition on direct summands, or

(ii) M is noetherian or artinian, or

(iii) M is finitely cogenerated, or

(iv) R is a noetherian ring, or

(v) R is a semiperfect ring.

Then M is an \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module if and only if M is semisimple.

Proof

Applying [3, Proposition 10.14] and [18, 21.3(2), 41.6 and 42.6], we conclude that M is a sum of indecomposable submodules. Now the result follows from Corollary 2.19. \(\square \)

Proposition 2.21

Let M be a finitely generated R-module with essential socle. Then the following statements are equivalent:

(i) M is an \({\mathfrak {s}}\)-coseparable module;

(ii) \(Rad(M)=0\);

(iii) Every simple submodule of M is a direct summand of M.

Proof

(i) \(\Rightarrow \) (ii) By Theorem 2.7.

(ii) \(\Rightarrow \) (iii) This is immediate.

(iii) \(\Rightarrow \) (i) Let U be a nonzero submodule of M. Since Soc(M) is essential in M, U contains a simple submodule T. By (iii), T is a nonzero direct summand of M. This completes the proof. \(\square \)

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 \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-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 \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable.

Next, we focus on the question of when is the direct sum of two \({\mathfrak {s}}\)-coseparable modules, \({\mathfrak {s}}\)-coseparable?

Combining Theorem 2.7 and [1, Theorem 2.9], we obtain the following proposition.

Proposition 2.24

Any finite direct sum of \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable modules is \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable.

Proposition 2.25

Let \(M=M_1 \oplus M_2\) be a direct sum of \({\mathfrak {s}}\)-coseparable submodules \(M_1\) and \(M_2\) such that \(M_2\) is finitely generated and semisimple. Then M is \({\mathfrak {s}}\)-coseparable.

Proof

Let U be a nonzero cofinite submodule of M. Then

$$\begin{aligned} M/U = ((M_1+U)/U) + ((M_2+U)/U). \end{aligned}$$

Since \((M_2+U)/U\) is semisimple, it follows that \((M_1+U)/U\) is a direct summand of M/U. Hence, \(M_1/(U \cap M_1)\) is finitely generated.

Case 1: Assume that \(U \cap M_1 \ne 0\). Since \(M_1\) is \({\mathfrak {s}}\)-coseparable, there exists a nonzero direct summand \(U_1\) of \(M_1\) such that \(U_1 \subseteq U \cap M_1 \subseteq U\) 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 \(U\cap M_1=0\). Then \(M_1\) is finitely generated. Using Proposition 2.24, it follows that U contains a nonzero cofinite direct summand of M.

Consequently, M is \({\mathfrak {s}}\)-coseparable. \(\square \)

Corollary 2.26

Let R be a right noetherian ring. Then any finite direct sum of \({\mathfrak {s}}\)-coseparable R-modules is \({\mathfrak {s}}\)-coseparable.

Proof

Let \(M=M_1\oplus M_2\) be a direct sum of two \({\mathfrak {s}}\)-coseparable submodules.

Case 1: Assume that \(M_1\) is finitely generated. Then \(M_1\) is semisimple by Corollary 2.20. Applying Proposition 2.25, we conclude that M is \({\mathfrak {s}}\)-coseparable.

Case 2: Assume that both \(M_1\) and \(M_2\) are not finitely generated. Then M is \({\mathfrak {s}}\)-coseparable by [8, Corollary 3.6] and Theorem 2.7. \(\square \)

A module M is called a duo module if for every submodule N of M and any endomorphism f of M, we have \(f(N) \subseteq N\).

Proposition 2.27

Let \(M=\oplus _{i\in I} M_i\) be a direct sum of nonzero submodules \(M_i\) \((i\in I)\) such that for every submodule N of M, we have \(N=\oplus _{i\in I}(N \cap M_i)\) (e.g., M is a duo module). Assume that \(M_i\) is \({\mathfrak {s}}\)-coseparable for all \(i \in I\). Then M is \({\mathfrak {s}}\)-coseparable.

Proof

Case 1: Assume that M is finitely generated. Then I is a finite set and each \(M_i\) (\(i \in I\)) is finitely generated. Using Proposition 2.24, we see that M is \({\mathfrak {s}}\)-coseparable.

Case 2: Assume that M is not finitely generated. Note that M is coseparable by [8, Proposition 3.16]. Hence, M is \({\mathfrak {s}}\)-coseparable by Theorem 2.7. \(\square \)

Completely \({\mathfrak {s}}\)-coseparable modules

Definition 3.1

A module M is called completely \({\mathfrak {s}}\) -coseparable if every direct summand of M is \({\mathfrak {s}}\)-coseparable.

It is clear that semisimple modules are completely \({\mathfrak {s}}\)-coseparable. In addition, every module M with \(Rad(M)=M\) is completely \({\mathfrak {s}}\)-coseparable.

Example 3.2

(i) From Proposition 2.16(ii), it follows that every \({\mathfrak {f}}{\mathfrak {g}}{\mathfrak {s}}\)-coseparable module is completely \({\mathfrak {s}}\)-coseparable.

(ii) By Example 2.12, the \({\mathbb {Z}}\)-module \(M={\mathbb {Z}}^{({\mathbb {N}})}\) is \({\mathfrak {s}}\)-coseparable. On the other hand, M is not completely \({\mathfrak {s}}\)-coseparable since \({\mathbb {Z}}\) is not \({\mathfrak {s}}\)-coseparable by Proposition 2.5.

In the next proposition, we provide more examples of completely \({\mathfrak {s}}\)-coseparable modules.

Proposition 3.3

Let \(M=M_1\oplus M_2\) be a direct sum of a semisimple submodule \(M_1\) and a submodule \(M_2\) with \(Rad(M_2)=M_2\). Then, M is completely \({\mathfrak {s}}\)-coseparable.

Proof

If \(M_2=0\), it is clear that M is completely \({\mathfrak {s}}\)-coseparable. Now assume that \(M_2 \ne 0\). Then, M is not finitely generated. Using Theorem 2.7, the proof is completed by showing that M is coseparable. Let U be a nonzero submodule of M with \(M/U = ((M_1+U)/U) + ((M_2+U)/U)\) finitely generated. Since \((M_1+U)/U\) is semisimple, it follows that \((M_2+U)/U\) is a direct summand of M/U. Hence, \((M_2+U)/U\) is finitely generated. But \(Rad((M_2+U)/U) = (M_2+U)/U\). Therefore, \((M_2+U)/U = 0\). This yields \(M_2 \subseteq U\). By modularity, we have \(U=(U \cap M_1) \oplus M_2\). Since \(U \cap M_1\) is a direct summand of \(M_1\), it follows that U is a direct summand of M. This completes the proof. \(\square \)

Recall that a module M is said to have finite uniform dimension if M does not contain an infinite independent set of submodules. Dually, a module M is said to have finite hollow dimension if M does not contain an infinite coindependent family of submodules; that is, for some \(n \in {\mathbb {N}}\), there exists an epimorphism from M to a direct sum of n nonzero modules but no epimorphism from M to a direct sum of more than n nonzero modules (see, for example, [6, p. 47]). Next, we present some special cases where the converse of Proposition 3.3 holds.

Corollary 3.4

Let R be a right noetherian ring and let M be an R-module with \(S = End_R(M)\). Assume that one of the following conditions is fulfilled:

(i) The ring S is semilocal;

(ii) The module \(S_S\) has finite uniform dimension;

(iii) The module M has finite hollow dimension;

(iv) The module M has finite uniform dimension;

(v) The module M is artinian.

Then, M is completely \({\mathfrak {s}}\)-coseparable if and only if \(M=M_1 \oplus M_2\) is a direct sum of submodules \(M_1\) and \(M_2\) such that \(Rad(M_1)=M_1\) and \(M_2\) is finitely generated and semisimple.

Proof

Note that M satisfies ACC on direct summands by [8, Poof of Proposition 3.13].

\((\Rightarrow )\) By [8, Proposition 3.13], \(M=M_1 \oplus M_2\) such that \(Rad(M_1)=M_1\) and \(M_2\) is finitely generated. Since M is completely \({\mathfrak {s}}\)-coseparable, \(M_2\) is \({\mathfrak {s}}\)-coseparable. Clearly, \(M_2\) has ACC on direct summands. Hence, \(M_2\) is semisimple by Corollary 2.20.

\((\Leftarrow )\) By Proposition 3.3. \(\square \)

Proposition 3.5

Let R be a semiperfect ring and let M be a projective R-module. Then, M is completely \({\mathfrak {s}}\)-coseparable if and only if M is semisimple.

Proof

The sufficiency follows by Proposition 3.3. For the necessity, suppose that M is completely \({\mathfrak {s}}\)-coseparable. By [3, Theorem 27.12], M has a direct decomposition \(M=\oplus _{i\in I}M_i\) such that each \(M_i\) \((i \in I)\) is indecomposable. Therefore, \(M_i\) is \({\mathfrak {s}}\)-coseparable and projective for each \(i\in I\). Note that \(Rad(M_i) \ne M_i\) for each \(i\in I\) (see [3, Proposition 17.14]). Using Proposition 2.5, it follows that each \(M_i\) \((i \in I)\) is a simple submodule of M. Therefore, M is semisimple. \(\square \)

Proposition 3.6

Let R be a right noetherian ring and let M be an injective R-module. Then, M is completely \({\mathfrak {s}}\)-coseparable if and only if \(M=M_1 \oplus M_2\) is a direct sum of submodules \(M_1\) and \(M_2\) such that \(Rad(M_1)=M_1\) and \(M_2\) is semisimple.

Proof

The sufficiency is obvious. Conversely, suppose that M is completely \({\mathfrak {s}}\)-coseparable. By [3, Theorem 25.6], \(M=\oplus _{i\in I}M_i\) is a direct sum of indecomposable submodules \(M_i\) \((i \in I)\). It follows that each \(M_i\) \((i\in I)\) is \({\mathfrak {s}}\)-coseparable. By Proposition 2.5, for each \(i \in I\), we have \(Rad(M_i)=M_i\) or \(M_i\) is simple. The result follows. \(\square \)

A ring R is said to be an SSI-ring if every semisimple right R-module is injective.

Proposition 3.7

Let R be a ring. Then, the following conditions are equivalent:

(i) Every completely \({\mathfrak {s}}\)-coseparable R-module is injective;

(ii) R is a right noetherian right V-ring.

Moreover, in this case, every completely \({\mathfrak {s}}\)-coseparable R-module is semisimple.

Proof

(i) \(\Rightarrow \) (ii) Let M be a semisimple module. Then, M is completely \({\mathfrak {s}}\)-coseparable. By hypothesis, M is injective. Hence, R is an SSI-ring. Thus R is a right noetherian right V-ring by [5, Proposition 1].

(ii) \(\Rightarrow \) (i) Let M be a completely \({\mathfrak {s}}\)-coseparable module. By [5, Proposition 1], Soc(M) is injective. Then, \(M=Soc(M) \oplus N\) for some submodule N of M with \(Soc(N)=0\). Suppose \(N\ne 0\). As R is a right V-ring, N contains a maximal submodule L. Since N is \({\mathfrak {s}}\)-coseparable, \(N=U \oplus K\) is a direct sum of submodules U and K such that \(U \subseteq L\) and K is a nonzero finitely generated submodule of M. Note that K is \({\mathfrak {s}}\)-coseparable. As R is a right noetherian ring, K is semisimple by Corollary 2.20. This implies that \(Soc(N) \ne 0\), a contradiction. It follows that \(N=0\) and \(M=Soc(M)\) is injective. \(\square \)

Corollary 3.8

The following assertions are equivalent for a commutative ring R:

(i) Every completely \({\mathfrak {s}}\)-coseparable R-module is injective;

(ii) R is a semisimple ring.

Proof

This is a consequence of [5, p. 236 Proposition 1 and Corollary] and Proposition 3.7. \(\square \)

Next, we present a special subclass of completely \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable.

Example 3.9

(i) Let R be a commutative noetherian ring and let \(\lbrace U_\lambda \rbrace _{\lambda \in \Lambda } \) be a collection of simple R-modules. Then, \(M=\prod _ {\lambda \in \Lambda } U_\lambda \) is an md-module by [4, Theorem 6.2]. This implies that M is completely \({\mathfrak {s}}\)-coseparable.

(ii) Let a module \(M=M_1 \oplus M_2\) be a direct sum of a submodule \(M_1\) with \(Rad(M_1)=M_1\) and a semisimple submodule \(M_2\). From the proof of Proposition 3.3, it follows that M is an md-module.

Example 3.10

Let M be a regular R-module over a right noetherian ring R. Let U be a nonzero submodule of M with M/U finitely generated. Then, \(M=U+K\) for some finitely generated submodule K of M. Since R is right noetherian, \(U\cap K\) is a finitely generated submodule of M. But M is regular. Then, \(U\cap K\) is a direct summand of M. Therefore, \(U\cap K\) is a direct summand of K. Hence, \(K=(U\cap K) \oplus L\) for some submodule L of K. Clearly, \(M=U\oplus L\). Hence, M is an md-module. Therefore, M is completely \({\mathfrak {s}}\)-coseparable.

Rings whose modules are \({\mathfrak {s}}\)-coseparable

In this section, we characterize some classes of rings via \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable.

Theorem 4.1

The following statements are equivalent for a ring R:

(i) The R-module \(R_R\) is \({\mathfrak {s}}\)-coseparable;

(ii) R is a weakly regular semiprimitive ring;

(iii) For every \(0 \not = a \in R\), there is \(0 \not = x \in R\) with \(x = xax\);

(iv) Every (finitely generated) free R-module F is \({\mathfrak {s}}\)-coseparable;

(v) Every finitely generated projective R-module is \({\mathfrak {s}}\)-coseparable;

(vi) For every finitely generated projective R-module P, \(End_R(P)\) is a weakly regular semiprimitive ring.

Proof

(i) \(\Leftrightarrow \) (ii) This follows from Theorem 2.7.

(ii) \(\Leftrightarrow \) (iii) By [16, Remark 15.3(3)].

(i) \(\Rightarrow \) (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 \({\mathfrak {s}}\)-coseparable by Example 2.11(i). Now if F is finitely generated, then F is \({\mathfrak {s}}\)-coseparable by Proposition 2.24.

(iv) \(\Rightarrow \) (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 \({\mathfrak {s}}\)-coseparable by Proposition 2.16.

(v) \(\Rightarrow \) (i) This is immediate.

(ii) \(\Rightarrow \) (vi) Let P be a finitely generated projective R-module. From [10, Theorem 3.8], it follows that \(S=End_R(P)\) is a weakly regular ring. Moreover, by [3, Corollary 17.12], we have \(Rad(S)=Hom_R(P, PRad(R))=0\). Thus S is a semiprimitive ring.

(vi) \(\Rightarrow \) (ii) This follows from the fact that the rings \(End_R(R_R)\) and R are isomorphic. \(\square \)

The next corollary follows directly from Theorem 4.1.

Corollary 4.2

For any ring R, the right R-module \(R_R\) is \({\mathfrak {s}}\)-coseparable if and only if it is completely \({\mathfrak {s}}\)-coseparable.

Next, we shed some light on the class of rings R for which every cyclic R-module is \({\mathfrak {s}}\)-coseparable.

Proposition 4.3

Let R be a ring such that every cyclic R-module is \({\mathfrak {s}}\)-coseparable. Then, R is a right V-ring.

Proof

Let I be a right ideal of R. By hypothesis, R/I is an \({\mathfrak {s}}\)-coseparable R-module. Hence, \(Rad(R/I)=0\) by Theorem 2.7. Therefore, R is a right V-ring by [13, Theorem 3.75]. \(\square \)

Corollary 4.4

The following statements are equivalent for a commutative ring R:

(i) Every cyclic R-module is \({\mathfrak {s}}\)-coseparable;

(ii) R is a von Neumann regular ring.

Proof

(i) \(\Rightarrow \) (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) \(\Rightarrow \) (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 \({\mathfrak {s}}\)-coseparable R-module. \(\square \)

It is shown in Example 2.12 that the class of \({\mathfrak {s}}\)-coseparable modules is not closed under factor modules. In the same vein, the next example shows that a factor module of a cyclic \({\mathfrak {s}}\)-coseparable module need not be \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable. On the other hand, using Corollary 4.4, it follows that the ring R has a cyclic R-module M which is not \({\mathfrak {s}}\)-coseparable. It is clear that \(M \cong R/I\) 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.

Corollary 4.6

Let R be a right semiartinian ring. Then, the following conditions are equivalent:

(i) R is a right V-ring;

(ii) Every finitely generated R-module is \({\mathfrak {s}}\)-coseparable;

(iii) Every cyclic R-module is \({\mathfrak {s}}\)-coseparable.

Proof

(i) \(\Rightarrow \) (ii) Let M be a finitely generated R-module. Then, \(Rad(M)=0\) since R is a right V-ring. Moreover, M is a semiartinian R-module by [15, p. 183 Proposition VIII.2.5]. Therefore, M is \({\mathfrak {s}}\)-coseparable by Corollary 2.22.

(ii) \(\Rightarrow \) (iii) This is clear.

(iii) \(\Rightarrow \) (i) This follows from Proposition 4.3. \(\square \)

The next result provides some information about the class of rings R for which every R-module is \({\mathfrak {s}}\)-coseparable.

Corollary 4.7

The following statements are equivalent for a ring R:

(i) Every R-module is \({\mathfrak {s}}\)-coseparable;

(ii) R is a weakly regular semiprimitive ring such that every factor module of an \({\mathfrak {s}}\)-coseparable module is also \({\mathfrak {s}}\)-coseparable.

Proof

(i) \(\Rightarrow \) (ii) By Theorem 4.1.

(ii) \(\Rightarrow \) (i) Let M be an R-module. Then, \(M \cong F/N\) for some free R-module F and a submodule N of F. By Theorem 4.1, F is \({\mathfrak {s}}\)-coseparable. By hypothesis, M is also an \({\mathfrak {s}}\)-coseparable module. \(\square \)

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 \({\mathfrak {s}}\)-coseparable. For example, the \({\mathbb {Z}}\)-module \({\mathbb {Z}}\) is not \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable if and only if R is a weakly regular semiprimitive ring.

Proof

(\(\Rightarrow \)) This follows from Theorem 4.1.

(\(\Leftarrow \)) By Theorem 4.1, every finitely generated projective module is \({\mathfrak {s}}\)-coseparable. Now let P be a projective module which is not finitely generated. Applying [14, Corollary 5.5] and [17, Theorem 1], it follows that P is a direct sum of finitely generated submodules. Moreover, we have \(Rad(P)=0\) since \(Rad(R)=0\) (see [3, Proposition 17.10]). Therefore, P is \({\mathfrak {s}}\)-coseparable by Proposition 2.10. \(\square \)

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 \({\mathfrak {s}}\)-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 \(Rad(E(S)) \ne E(S)\) 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 \({\mathfrak {s}}\)-coseparable;

(ii) E(S) is \({\mathfrak {s}}\)-coseparable for every simple R-module S;

(iii) R is a right V-ring.

Proof

(i) \(\Rightarrow \) (ii) This is immediate.

(ii) \(\Rightarrow \) (iii) Let S be a simple R-module. Since E(S) is an indecomposable \({\mathfrak {s}}\)-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) \(\Rightarrow \) (i) Since R is a right V-ring, every finitely cogenerated R-module is semisimple. The result follows. \(\square \)

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 \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-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 \({\mathfrak {s}}\)-coseparable;

(ii) Every nonzero R-module contains a nonzero finitely generated injective submodule;

(iii) R is a right semiartinian right V-ring.

Then, (i) \(\Rightarrow \) (ii) \(\Rightarrow \) (iii) hold.

Proof

(i) \(\Rightarrow \) (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 \({\mathfrak {s}}\)-coseparable. Then, there exist submodules N and K of E(M) such that \(E(M)=N\oplus K\), \(N \subseteq L\) and K is a nonzero finitely generated submodule of M. But M is essential in E(M). Therefore, \(M\cap K \ne 0\). As K is injective, K is \({\mathfrak {s}}\)-coseparable. Hence, \(M\cap K\) contains a nonzero direct summand U of K. It is clear that U is a finitely generated injective submodule of M.

(ii) \(\Rightarrow \) (iii) By [7, Theorem 13]. \(\square \)

Corollary 4.13

Let R be a right perfect ring. Then, the following statements are equivalent:

(i) Every injective R-module is \({\mathfrak {s}}\)-coseparable;

(ii) R is a semisimple ring.

Proof

(i) \(\Rightarrow \) (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.

(ii) \(\Rightarrow \) (i) This is obvious. \(\square \)

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Ech-chaouy, R., Idelhadj, A. & Tribak, R. On strongly coseparable modules. Arab. J. Math. (2021). https://doi.org/10.1007/s40065-021-00333-1

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