Generalization of Certain Characteristics of Multiplication Modules to Locally Multiplication Modules

Abstract

Background: Consider a commutative ring  with a non-zero identity and an module . A multiplication module is an module that satisfies the multiplication property. That is, every submodule  is presented as the product of an ideal  of the ring and the module  itself.

Materials and Methods: An module  is a locally multiplication module if or  is a multiplication module for each prime ideal  of . In this paper, certain properties of multiplication modules are extended to locally multiplication modules by providing conditions under which this generalization is possible.

Results: A locally multiplication module  is Noetherian whenever the ring is both semi-local and satisfies the conditions of an ascending chain on ideals with prime/semiprime property.  As well as, if  is a faithful locally multiplication module and  is a prime ideal with , then a bijection exists between ideals and submodules of  and  .  Also, if  is a locally multiplication module and  as the submodule of  is both prime and primary whenever the set of  contained in , along with several other results.   

Conclusion: Various conditions are presented in terms of faithfulness, cyclic, prime, semi-prime, primary, and others, to illustrate the generalization of certain properties of multiplication modules to locally multiplication modules.