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

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  to  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. ./files/site1/files/51/%D9%86%DB%8C%DA%A9_%D8%B3%D8%B1%D8%B4%D8%AA.pdf