Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules

Nikseresht, Ashkan

doi:10.29252/mmr.5.1.107

Volume 5, Issue 1 (Vol. 5, No. 1 2019) mmr 2019, 5(1): 107-120 | Back to browse issues page

‎ 10.29252/mmr.5.1.107

‎ 20.1001.1.25882546.1398.5.1.8.0

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Nikseresht A. Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules. mmr 2019; 5 (1) :107-120
URL: http://mmr.khu.ac.ir/article-1-2627-en.html

Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules

Ashkan Nikseresht ^*

Shiraz University , ashkan_nikseresht@yahoo.com

Abstract: (2350 Views)

Introduction
Suppose that

is a commutative ring with identity,

is a unitary

-module and

is a multiplicatively closed subset of

.
Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was generalized to commutative rings with zero-divisors and to modules. Also recently, the factorization properties of an element of a module with respect to a multiplicatively closed subset of the ring has been investigated. It has been shown that using these general views, one can derive new results and insights on the classic case of factorization theory in integral domains.
An important and attractive question in this theory is understanding how factorization properties of a ring or a module behave under localization. In particular, Anderson, et al in 1992 showed that if

is an integral domain and every principal ideal of

contracts to a principal ideal of

, then there are strong relations between factorization properties of

and

. In the same paper and also in another paper by Aḡargün, et al in 2001 the concepts of inert and weakly inert extensions of rings were introduced and the relation of factorization properties of

and

, under the assumption that

is (weakly) inert, is studied.
In this paper, we generalize the above concepts to modules and with respect to a multiplicatively closed subset. Then we utilize them to relate the factorization properties of

and

.
Material and methods
We first recall the concepts of factorization theory in modules with respect to a multiplicatively closed subset of the ring. Then, we define multiplicatively closed subsets conserving cyclic submodules of

and say that

conserves cyclic submodules of

, when the contraction of every cyclic submodule of

is a cyclic submodule. We present conditions on

equivalent to conserving cyclic submodules of

and study how factorization properties of

is related to those of

, when

coserves cyclic submodules of

Finally we present generalizations of inert and weakly inert extensions of rings to modules and investigate how factorization properties behave under localization with respect to

, when

is inert or weakly inert.

Results and discussion
We show that if

is an integral domain,

is torsion-free and

conserves cyclic submodules of

, then

splits

(as defined by Nikseresht in 2018) and hence factorization properties of

and those of

are strongly related. Also we show that under certain conditions, the converse is also true, that is, if

splits

, then

conserves cyclic submodules of

.
Suppose that

is a multiplicatively closed subset of

containing

and

. We show that if

is a

-weakly inert extension, then there is a strong relationship between

- factorization properties of

and

-factorization properties of

. For example, under the above assumptions, if

is also torsion-free and has unique (or finite or bounded) factorization with respect to

, then

has the same property with respect to

.
Conclusion
In this paper, the concepts of a multiplicatively closed subset conserving cyclic submodules and inert and weakly inert extensions of modules are introduced and utilized to derive relations between factorization properties of a module

and those of its localization

. It is seen that many properties can be delivered from one to another when

conserves cyclic submodules or when

is a weakly inert extension, especially when

is an integral domain and

is torsion-free.
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Keywords: Multiplicatively closed subsets conserving cyclic submodules, Inert extension, Atomic module, Unique factorization module

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