Modules with Chain Conditions on S-Closed Submodules

Submodules of Secondary Modules

On the Prime Submodules of Multiplication Modules

By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results. 1. Introduction. Let R b...

Modules Satisfying the Ascending Chain Condition on Submodules with a Bounded Number of Generators

It is proved that if R is a right and left Noetherian ring then the right R-module RI satisfies the ascending chain condition on n-generated submodules, for every positive integer n. Mathematics Subject Classification (2000): 16P70

Pure Submodules of Multiplication Modules

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

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 gener...