A generalization of purely extending modules relative to a torsion theory

In this work we introduce a new concept, namely, τs-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring R with unit, if R is purely τs-extending then every cyclic τ -nonsingular R-module is flat and we show that this fact is true over a principal ideal domain as well. Also, we make a classification for the direct sums of the rings to be purely τs-extending.


Introduction
Injective modules have been intensively studied in the 1960s and 1970s in module theory and more generally in algebra. As a generalization of injective modules extending modules (CS), that is every closed submodule is a direct summand, have been studied widely in last three decades. In general setting Chatters and Hajarnavis [6], Harmancı and Smith [21], Kamal and Muller [22] and their schools can be mentioned involving studies of extending modules.
Recently, torsion-theoretic analogues of extending modules has been an interest to extend many results and concepts from extending modules to a torsion theory such primarily studies as Asgari and Haghany [3], Berktaş, Dogruöz and Tarhan [5], Crivei [10], Ç eken and Alkan [11], Doǧruöz [12]. Clark [7] defined a module M is purely extending if every submodule of M is essential in a pure submodule of M , equivalently every closed submodule of M is pure in M . Al-Bahrani [1] generalized purely extending modules as a purely y-extending module using s-closed submodules which was defined by Goodearl [19] such as a submodule N of a module M is s-closed in M if M/N is non-singular. So a module M is called purely y-extending if every s-closed submodule of M is pure in M . In fact Al-Bahrani [1] belike misused the terminology of s-closed submodules. They used the term y-closed (purely y-extending) instead of s-closed (purely s-extending) respectively. In this study we use s-closed submodule and purely s-extending module instead of y-closed submodule and purely y-extending module in the sense of Al-Bahrani [1].
We use the concept 'purity' in the sense of Cohn [9] (as in [7]) which implies definition of Anderson and Fuller [2], that is, a submodule N of an R-module M is called pure submodule in M in case IN = N ∩ IM for each finitely generated right ideal I of the ring R (see also [23] ). In the present paper we introduce purely τ s -extending modules and then we extend many results from [1], [7] and [19] to this new concept.
For instance we show that: Theorem 18: Let R be a τ -torsion ring and M be an R-module. Let E(M ) be an injective hull of M . Then M is a purely τ s -extending module if and only if Proposition 26: Let R be a ring with identity. If R R is purely τ s -extending then every cyclic τ -nonsingular R-module is flat. and Theorem 31: Let R be a commutative integral domain. Then the following properties are equivalent: : for each n ∈ N, n R is an extending module. (7): for each n ∈ N, n R is a purely extending module. (8): for each n ∈ N, n R is a purely s-extending module. (9): for each n ∈ N, n R is a purely τ s -extending module.
Throughout the work R will be an associative ring with identity and all R-modules will be unitary left R-modules unless otherwise stated. R−M od will be the category of unitary left R-modules, and all modules and module homomorphisms will belong to R − M od. Let τ : (T , F ) be a torsion theory on R − M od. The modules in T are called τ -torsion modules and the modules in F are called τ -torsion-free modules. Let M ∈ R − M od. Then the τ -torsion submodules of M , denoted by τ (M ), is defined to be the sum of all τ -torsion submodules of M . Thus τ (M ) is the unique largest τ -torsion submodule of M and τ (M/τ (M )) = 0 for an R-module M . The torsion class T is given T := {M ∈ R − M od | τ (M ) = M } and F is refered to as torsion-free class and given by F := {M ∈ M od − R | τ (M ) = 0}. In our study τ will be a hereditary torsion theory on R-Mod and we mean R is τ -torsion ring if then M is called τ -singular module and if Z τ (M ) = 0 then M is called τ -nonsingular module ([18]).
For elementary, additional and unexplained terminology the reader is referred to [2] or [27] for module and ring theory, [17] and [25] for torsion theory, [13] for extending modules and [23] for homological algebra.

Purely τ s -Extending Modules
Let M be an R-module and N be a submodule of M . We call N is τ s -closed submodule of M if the factor module M/N is a τ -nonsingular and it is denoted by N ≤ τsc M . Definition 1. Let M be an R-module. If every τ s -closed submodule of M is pure in M then we call M is a purely τ s -extending module. It is denoted briefly p-τ s -extending.
Proof. Let N be a τ s -closed submodule of M . Then the factor module M/N is τ -nonsingular i.e., Z τ (M/N ) = 0. Since R is τ -torsion, Z τ (M/N ) = Z(M/N ). Assume N is not closed in M . Then there exists a submodule K of M such that K contains N as an essential submodule. So the factor module K/N is singular [19]. Hence Z(K/N ) = K/N . On the other hand since Z(K/N ) is a submodule of Z(M/N ), we have Z(K/N ) = 0. Hence K/N is nonsingular. But since K/N is singular, it must be zero (i.e K/N = 0). Therefore N = K and so N is closed submodule of M . As in general extending module theory we have some of the fundamental properties of purely τ s -extending modules as follows: Similarly it can be shown that M 2 is also purely τ s -extending module.
Proof. It is clear from Lemma 4.

Lemma 7. The class of τ -nonsingular modules is closed under extensions by short exact sequences.
Proof. Let C and A be τ -nonsingular modules and consider the following short exact sequence For every τ -singular R-module M , using Lemma 6, we have Hom R (M, C) = 0 and Hom R (M, A) = 0. Then the following short exact sequence yields Hom R (M, B) = 0. Again by Lemma 6 the R-module B must be τ -nonsingular.
Next we can show τ s -closed submodules have transitivity property.
We must show that Z τ (M/K) = 0. Consider the following short exact sequence By Lemma 7, the class of τ -nonsingular modules are closed under extensions by short exact sequences. Since Now we have some basic properties as follows.
Lemma 9. Any τ s -closed submodule of a purely τ s -extending module is purely τ s -extending.
Proof. Let M be a purely τ s -extending module and let N be a τ s -closed submodule of M . Then M/N is τ -nonsingular. Let K be a τ s -closed submodule of N . Then by Lemma 8, K is a τ s -closed submodule of M . Since M is purely τ s -extending module, K is pure in M . By [15, Proposition 1.2 (2)], K is pure in N . So N is purely τ s -extending module.
There exist submodules K, L of a module M such that K and L both closed submodules of M but K ∩ L is not closed in K, L or M (see [19,Example 1.6]). But we have the following in our case. Proof. Let A be a direct summand of E(M ) such that A ∩ M is τ s -closed in M . Consider the following short exact sequences of R-modules Thus A is also a direct summand of A + M . Then the short exact sequence

Purely τ s -Extending Rings
If the ring R is purely τ s -extending as an R-module over itself then R is called purely τ s -extending. Proof. Let R R be a purely τ s -extending module. Let M = Ra be a cyclic τ -nonsingular R-module which is generated by a where a ∈ R. Define the map f : R → M with f (r) = ra. Clearly f is an epimorphism and Ker(f ) = Ann(a). So R/Ker(f ) = R/Ann(a) ∼ = Ra. Moreover, since Ra is a τ -nonsingular module and the class of τ -nonsingular modules is closed under isomorphisms R/Ann(a) is τ -nonsingular. Hence Ann(a) is τ s -closed in R. By the hypothesis Ann(a) is pure in R. Since R is flat and Ann(a) is pure in R, R/Ann(a) is flat by [2,Lemma 19.18]. Therefore Ra is flat.
Proposition 27. Let R be a principal ideal domain (for short PID). If every cyclic τ -nonsingular R-module is flat then R R is purely τ s -extending.
Proof. Let K be a τ s -closed ideal of R. Then R/K is τ -nonsingular. Since R is P ID the factor ring R/K is also P ID. Hence R/K is cyclic. By hypothesis R/K is flat. Thus by [2,Lemma 19.18], K is pure in R. Then R is purely τ s -extending. When the ring R is semi-hereditary we have Proposition 26 with its converse also as follows.
Theorem 28. Let R be a semi-hereditary ring. Then R  Conversely let C be a τ s -closed ideal of R. Then R/C is τ -nonsingular and also R/C is cyclic. Hence by the hypothesis R/C is flat. By [15, Theorem 1.7] we have C is pure in R. Thus R R is a purely τ s -extending module. R is a semi-hereditary ring, R is flat. Because of the direct sum of flat modules is flat R ⊕ R is flat ( [19]). Thus by [15,Proposition 1.3 (3)], we have the R-module M is flat.
For the converse, let C be a τ s -closed submodule of R ⊕ R. Then (R ⊕ R)/C is τ -nonsingular. On the other hand since R ⊕ R is a 2-generated τ -nonsingular, R-module (R⊕R)/C is also 2-generated τ -nonsingular R-module. By the hypothesis (R ⊕ R)/C is flat. Then by [15,Theorem 1.7] we get C is pure in R ⊕ R. Thus R ⊕ R is purely τ s -extending.
Corollary 30. Let R be a left semi-hereditary ring and I be a finite index set. Then ⊕ I R is purely τ s -extending if and only if every τ -nonsingular I-generated R-module is flat. Now we can give the following generalized characterization of purely τ s -extending modules.
Theorem 31. Let R be a commutative integral domain. Then the following properties are equivalent: projective and so (R ⊕ R)/K is flat (see [23,Proposition 3.46]). By [15,Proposition 1.3] K is pure in R ⊕ R. Hence R ⊕ R is a purely τ s -extending module.
Example 32. Let Z be the ring of integers. Then Z is a purely τ s -extending Z-module over itself.
Proof. Since Z is a principal ideal domain (PID), every ideal of Z is free and so it is projective. Therefore Z is a hereditary ring. Moreover Z is a semi-hereditary ring. By Theorem 31 ((1) ⇒ (5)) we have Z ⊕ Z is purely τ s -extending. Additionally, by Lemma 4 since the direct summands of purely τ s -extending modules are purely τ s -extending, we have Z is a purely τ s -extending module.