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We study how to read from the ordinary quiver of a representation-finite string algebra, the minimal lower boundm≥ 1 such that themth power of the radical of its module category vanishes. We also show how to read the degree of any irreducible morphism in such a representation-finite string algebras considering strings. [ABSTRACT FROM PUBLISHER]

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding. [ABSTRACT FROM PUBLISHER]

We study deformations of Leibniz algebra morphisms over a commutative local algebra base with 1. We construct the associated deformation cohomology that controls deformations using the cochain complex defining the Leibniz cohomology. [ABSTRACT FROM AUTHOR]

In this note,we answer a question of Hong et al. (2003) by proving that if a is a monomorphism of a reduced ring R , and R is a-skew Armendariz, then R is a-rigid. [ABSTRACT FROM AUTHOR]