### Stable Clifford theory for divisorially graded rings.

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In [2], Fuchs and Viljoen introduced and classified the ${B}^{*}$-modules for a valuation ring R: an R-module M is a ${B}^{*}$-module if $Ex{t}_{R}^{1}(M,X)=0$ for each divisible module X and each torsion module X with bounded order. The concept of a ${B}^{*}$-module was extended to the setting of a torsion theory over an associative ring in [14]. In the present paper, we use categorical methods to investigate the ${B}^{*}$-modules for a group graded ring. Our most complete result (Theorem 4.10) characterizes ${B}^{*}$-modules for a strongly graded ring R...

An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if R is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.

Let $\mathcal{G}$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $\mathcal{G}$ is a precover class are given. The next section studies the $\mathcal{G}$-precovers which are $\mathcal{G}$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma $-torsionfree and $\sigma $-torsionfree $\sigma $-injective covers are presented.

We study the construction of new multiplication modules relative to a torsion theory $\tau $. As a consequence, $\tau $-finitely generated modules over a Dedekind domain are completely determined. We relate the relative multiplication modules to the distributive ones.

The structure theory of abelian $p$-groups does not depend on the properties of the ring of integers, in general. The substantial portion of this theory is based on the fact that a finitely generated $p$-group is a direct sum of cyclics. Given a hereditary torsion theory on the category $R$- of unitary left $R$-modules we can investigate torsionfree modules having the corresponding property for all torsionfree factor-modules (and a natural requirement concerning extensions of some homomorphisms). This paper...

Recently Rim and Teply [11] found a necessary condition for the existence of $\sigma $-torsionfree covers with respect to a given hereditary torsion theory for the category $R$-mod. This condition uses the class of $\sigma $-exact modules; i.e. the $\sigma $-torsionfree modules for which every its $\sigma $-torsionfree homomorphic image is $\sigma $-injective. In this note we shall show that the existence of $\sigma $-torsionfree covers implies the existence of $\sigma $-exact covers, and we shall investigate some sufficient conditions for the converse....

We study bialgebra structures on quiver coalgebras and monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed...

Let R be an associative (not necessarily commutative) ring with unit. The study of flat left R-modules permits to achieve homological characterizations for some kinds of rings (regular Von Neumann, hereditary). Colby investigated in [1] the rings with the property that every left R-module is embedded in a flat left R-module and called them left IF rings. These rings include regular and quasi-Frobenius rings. Another useful tool for the study of non-commutative rings is the classical localization,...

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