Isomorphism Theorems for Crossed Squares of Commutative Algebras

Abstract

The isomorphism theorems for crossed squares of commutative algebras, which arise when the crossed modules of algebras are given an extra dimension, are the main subject of this paper. The definition of crossed squares of commutative algebras is given in this context, encompassing ideas like the crossed square ideal, image, and quotient crossed squares, as well as the kernel for crossed square morphisms. The study discusses the way how isomorphism theorems are applied to these structures and offers detailed proofs for this framework. Moreover, some necessary concepts such as quotient crossed squares, which were not previously specified in these structures, are also presented, and some basic properties are examined. The study provides opportunities for possible generalization to a number of different structures, including crossed n-cubes.