Almost Maximal Integral Domains and Finitely Generated Modules

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...

Ring With All Finitely Generated Modules Retractable

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 →...

On Reductions of Finitely Generated Ideals in Integral Domains

√(f1,...,fd+1)R[X] [3,p.124]. The question is whether an ideal (f1,...,fd+1) R[X] can be chosen as a reduction of I. We only know the following case of affine domains, which was developed by G. Lyubeznik [4]: Let R be an n-dimensional affine domain over an infinite field k and let I be an ideal of R. Then I has a reduction generated by n+1 elements. He also posed the following conjecture: Let A...