پژوهشهای ریاضی (Aug 2019)
Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules
Abstract
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
