FINITELY GENERATED G′ AND POSITIVE LAWS

K g is not finitely generated

Let Σg be a closed orientable surface of genus g. The mapping class group Modg of Σg is defined to be the group of isotopy classes of orientationpreserving diffeomorphisms Σg → Σg. Recall that an essential simple closed curve γ in Σg is called a bounding curve, or separating curve, if it is nullhomologous in Σg or, equivalently, if γ separates Σg into two connected components. Let Kg denote the...

Ring With All Finitely Generated Modules Retractable

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...

Presentations of Finitely Generated Submonoids of Finitely Generated Commutative Monoids

We give an algorithmic method for computing a presentation of any finitely generated submonoid of a finitely generated commutative monoid. We use this method also for calculating the intersection of two congruences on Np and for deciding whether or not a given finitely generated commutative monoid is t-torsion free and/or separative. The last section is devoted to the resolution of some simple ...