Modules with the Closed Sum Property

A module M is said to have the closed intersection property (briefly CIP) if, the intersection of any two closed submodules of M is again closed [6]. In this paper we present the dual of the CIP, namely, M has the closed sum property (briefly CSP) for which the sum of any two closed submodules, so the submodule generated by their union, is a closed submodule, too. We investigate the concept of CSP. Basic properties and some relations of these modules are given.


Introduction
Throughout this paper R is a commutative ring with identity and every R-module is a unitary.Let M be an R-module and N be a submodule of M. N is called essential in M (briefly  if, N has no proper essential extensions inside M; that is the only solution of the relation In this case the submodule K is called closure of N [8].A module M is said to be extending(or C1-module) if, for every submodule N of M there exists a direct summand K of M such that K N e  .Equivalently, every closed submodule of M is a direct summand [11].An R-module M is said to have the summand intersection property (briefly SIP) if, the intersection of two summands is again a summand [5].Dually, an R-module M is said to have the summand sum property (briefly SSP) if, the sum of two summands is again a summand [2].This paper is structured in two sections: in the first section we introduce some general properties of modules with CSP.We prove that for an R-module M, if M has the CSP then M has the SSP.Also, we show that, if M has the CSP then for all Rhomomorphism T S f  : where M T S  , and then Imf is closed in M, while in the second section we investigate some relationships between the concept of modules with CSP and other modules such as, modules with CIP and SIP.

Some basic properties of modules with the closed sum property.
In this section we introduce the concept of modules with CSP as a dual of modules with CIP.We investigate the basic properties of this type of modules.Before, we presented the following example.
This leads us to introduce the following.Proof.Assume that M has the CSP as R-module and are closed in M and hence

An injective R-module
In the following proposition we put certain condition under which ) (M E has CSP, where M is a module with CSP.Before, we consider the following condition for an R-module M.
. Again by [4], we get , by (*).Thus . □ Definition 1.6.We say that a module M has C ' 3 whenever L1 and L2 are closed submodules of M with 0 The following lemma was appeared in [1, Th.1.4.1.],we give the details of the proof. .Since M is an extending module, then A , B are direct summands of M and hence and hence . On the other hand, M has the C ' 3 , so by previous lemma and hence M has the SSP.□ Corollary 1.9.Let M be an R-module has the CSP.For any decomposition Alkan and Harmanci in [9] consider the following condition for a module M.

Proof. By previous proposition
Now we present the following condition.
The following gives a characterization for a module M with CSP.Proposition 1.13.Let M be an R-module.Then M has the CSP if and only if every closed submodule of M satisfies (⁑)c .

Proof. Let
, then A and B are closed in M and hence . Thus N satisfies (⁑)c.Conversely, let L1 and L2 are closed , so we have L1 and L2 are closed in K, thus by hypothesis , so (1) M has the SSP.
(3) M satisfies (⁑).Proof.Suppose A and B are closed submodules of The converse of prop.1.16 need not be true in general; as example: consider does not be have the CSP as Zmodule ( see Ex 1.1).Remark 1.17.Let M be an R-module.For any closed submodule K of M, if K M has the CSP then M has the CSP.

Proof. By taking
). 0 (  K □ Remark 1.18.Clearly, the Z-modules , it is not hard to prove that  is well-defined and homomorphism, then

Z Z 
as Z-module does not be have the CSP.This example shows the direct sum of two modules with the CSP need not be have the CSP.
The next proposition giving condition under which the direct sum of modules with the CSP has the CSP, too.Proof.Let A and B be two closed submodules of , then by the same way of the proof of [10, , by [8] .
. Then M has the CSP as R-module if and only if M has the CSP as  R module .
Proof.Obviously.□ Lemma 1.21.Let M be an R-module and let S be a multiplicative closed subset of R.
Proof.Let N be a closed submodule of M as R-module and assume We investigate the behavior of module with the CSP under localization.
Proposition 1.22.Let M be an R-module and let S be a multiplicative closed subset of R.
Then M has the CSP as R-module if and only if

Proof. It follows directly by above lemma. □
We end this section by the following corollary.
Corollary 1.23.Let M be an R-module.Then M has the CSP as R-module if and only if P M has the CSP as P R -module, for every maximal ideal P of R.

Modules with the CSP and CIP, and related concepts.
A module M is said to have strongly summand intersection property (briefly SSIP) if the intersection of any number of summands of M is again a direct summand of M. A module M is said to have strongly summand sum property (briefly SSSP) if the sum of any number of summands of M is again a direct summand of M. (D3) If L1 and L2 are summands of M with , We introduce the following condition (D ' 3) If K1 and K2 are closed submodules of M with , In this section we show that a module M has the CIP whenever M has the CSP and the D ' 3. Also, we prove that the concepts of quasi-continuous and the CSP are coincides whenever a module has the CIP.Moreover, many properties related with CSP and other modules are given in this section.
We begin with the following remark.
Remark 2.1.Every semisimple (Or simple) R-module has the CSP.Conversely, need not be true in general, as example; the Z-module Z has the CSP but it is not semisimple nor simple.(1) M has the SSIP.

Proof. Let
(2) M has the SIP.
,and hence by SSP on , this implies that 2 1 L L  is an injective submodules of M. Therefore by above proposition R is hereditary.

Proof
Im and hence M is divisible.□

Proof.
)  It follows directly by previous lemma.(1) R is semisimple.
(3) All R-modules has the CSP.(4) All projective R-modules has the CSP.

Definition 1 . 2 .Proposition 1 . 3 .
An R-module M is said to have the closed sum property (briefly CSP) if, the sum of any two closed submodules of M is again closed.Let M be an R-module.Then M has the CSP if and only if every closed submodule of M has the CSP.

Lemma 1 . 7 . 3 .
Let M be an R-module.Then M is a quasi-continuous module if and only if M has the C ' Proof.Let A and B are closed submodules of M with 0   B A closed in M and hence M has the CSP.□ Corollary 1.14.If M is an R-module has the CSP then M satisfies (⁑)c .Proof.Since M is closed, then the result it follows by above proposition.□ Corollary 1.15.Let M be an extending module.Then the following statements are equivalent.
the CSP as Z-module, to see this: Define

Proposition 1 .
19.Let M1 and M2 be two R-modules having the CSP such that

Lemma 2 . 2 .
Let M be a module has the CSP.If M has D ' 3 and M L   , then L has D ' 3.

Corollary 2 . 3 . 3 . 2 . 4 . 3 .Proposition 2 . 5 .Proposition 2 . 6 .
Let M be a module has the CSP.Then M has D ' 3 if and only if every direct summand of M has the D ' Proposition Suppose that every closed submodule of a module M has D ' If M has the CSP then M has the CIP.Proof.Assume that A and B are two closed submodules of M. Since B Let M be an R-module has D3.If M has the CSP then M has the CIP.Proof.By prop.1.8 M has the SSP, but M has D3, so by [9, lemma 19] M has the SIP.On the other hand, the CSP implies C1, thus by[2] M has the CIP.□ Let M be an R-module has the CSP.Then the following statements are equivalent.
it follows by cor.1.12.□ Proposition 2.18.Let M be a faithful and closed simple module over integral domain R.If M M  has the CSP then M is divisible.

Corollary 2 .Theorem 2 . 1 M
19.Let M be a torsion free and closed simple module over integral domain R.If M M  has the CSP then M is injective.Proof.It follows by above proposition and [3, prop 2.7].□ We shall consider the following definition.Definition 2.20.Let M and N be an R-modules.M is called closed dual-Rickart relative to N (briefly c-d-Rickart) if, for any ) 21. Let M be an R-module with CSP. is c-d-Rickart relative to 2 of R-modules and N be an R-module.Then N has the d-Rickart relative to N if and only if i M is c-d-Rickart relative to N, where I is a finite set.

MDefinition 2 .
is c-d-Rickart relative to N, for all ) 25.An R-module M is said to have strongly closed sum property (briefly SCSP) if, the sum of any number of closed submodules of M is again closed in M. Proposition 2.26.Let R be an extending ring.Then the following statements are equivalent.
Thus N has the CSP.Conversely, follows by taking .Let M be an R-module.Then M has the CSP if and only if every direct summand of M has the CSP.
Let M be an injective ( Or quasi-injective) R-module.Then M has the SSP if and only if M has the CSP.
1.10.Let M be an R-module.If M has the CSP then for all □ Proposition 1.11.Let M be an extending module.If M has the SSP then M has the CSP.Proof.Suppose A and B are closed submodules of M. Since M is extending then A and B are direct summands of M, but M has the SSP, so c  . □ Corollary 1.12.