Let R be a commutative ring. A full additive subcategory C of R-modules is triangulated if whenever two terms of a short exact sequence belong to C, then so does the third term. In this note we give a classification of triangulated subcategories of finitely generated modules over a principal ideal domain. As a corollary we show that in the category of finitely generated modules over a PID, thic...

In this paper we introduce a generalization of M-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . We will use the structure of the radical of a module in  and get some suitable results about this class of modules. Also the relation between injective hull in  and this kind of modules will be investigated in this article.   For a module  we show...

in this paper, we state some results on product of operators with closed rangesand we solve the operator equation txs*- sx*t*= a in the general setting of theadjointable operators between hilbert c*-modules, when ts = 1. furthermore, by usingsome block operator matrix techniques, we nd explicit solution of the operator equationtxs*- sx*t*= a.

There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ idea...

The construction of free R-modules over a cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a cartesian closed topological category.

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

Definition 1.1. If X is a topological space with a sheaf of ring OX , an OX-module is a sheaf F of abelian groups on X, together with the structure of an OX(U)-module on each F (U), which is compatible with restriction, in the sense that if V ⊆ U , for any s ∈ F (U) and f ∈ OX(U) we have ρUV (fs) = ρUV (f) · ρUV (s) in F (V ). A morphism between OX -modules F and G is a morphism φ of the underl...

A ring R is called left GF-closed, if the class of all Gorenstein flat left Rmodules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension. In this paper, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein fl...