Finitely-valuedf-modules

Free modules, finitely-generated modules

The following definition is an example of defining things by mapping properties, that is, by the way the object relates to other objects, rather than by internal structure. The first proposition, which says that there is at most one such thing, is typical, as is its proof. Let R be a commutative ring with 1. Let S be a set. A free R-moduleM on generators S is an R-module M and a set map i : S →...

MULTIPLICATION MODULES THAT ARE FINITELY GENERATED

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...

A characterization of finitely generated multiplication modules

 Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...

Quasi-Exact Sequence and Finitely Presented Modules

The notion of quasi-exact sequence of modules was introduced by B. Davvaz and coauthors in 1999 as a generalization of the notion of exact sequence. In this paper we investigate further this notion. In particular, some interesting results concerning this concept and torsion functor are given.

Ring With All Finitely Generated Modules Retractable