Categorical aspect of crossed modules on associative algebra

This thesis devoted for detailed survey about categorical structure of associative algebras and their higher dimensional versions such as precrossed modules and crossed modules. The first chapter organized as preparation for the next chapters. Namely, we give the definitions of special notions products, co-products, pullbacks, pushouts, etc. In the second chapter we consider a particular case; associative algebras. The construction of these categorical notions in the case of associative algebras were examined in details. The last chapter consist of a survey about the crossed module theory in associative algebras. The notion was first introduced by Lue,[1] 1968 and extensively worked in [2]. In the recent works, the representability of actions in the category of (pre)crossed modules on associative algebras was reached , see[3]. In this chapter we construct the categorical notions in the case of crossed modules under certain conditions and give the construction of split extension classifiers in the category