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2 Colloquium Mathematicum

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Quiver bialgebras and monoidal categories

100%

Huang H.-L. , Torrecillas B.

Colloquium Mathematicum

2013

tom 131

nr 2

287-300

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 monoidal categories.

Weak Baer modules over graded rings

100%

Teply M. L. , Torrecillas B.

Colloquium Mathematicum

1998

tom 75

nr 1

19-31

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 $Ext^1_R(M,X)=0$ 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 over a finite group G with $|G|^{−1} \in R$. Motivated by the results of [8], [9], [10] and [15], we also study the condition that every non-singular R-module is a $B^∗$-module with respect to the Goldie torsion theory; for the case in which R is a strongly graded ring over a group, extensive information is obtained for group rings of abelian, solvable and polycyclic-by-finite groups.