Ideal and Factor Conditions for Crossed Modules and Algebra-Algebroids

Abstract

In this paper, using the equivalence between the category of crossed modules of algebras and the category of algebra-algebroids, we will explore the notions of ideality and factors for these algebraic structures. We give the structure of a two sided ideal of an algebra-algebroid and the notion of quotient algebra-algebroid. By considering a two sided ideal of an algebra-algebroid, we show that the crossed module corresponding to this ideal is a crossed ideal of the crossed module corresponding to the algebra-algebroid. Conversely, by taking a crossed ideal of a crossed module, we also show that the corresponding algebra-algebroid to this crossed ideal is an ideal of the algebra-algebroid corresponding to the crossed module.