Split (Pre)crossed Modules over $R$-Algebroids

Abstract

In this paper, we introduce (d-)split and (d-)split$^{+}$ epimorphisms and (d-)split and d-split$^{\ast}$ (pre)crossed modules in the context of algebroids. Moreover, we examine their categorical properties, and in particular, we give a necessary and sufficient condition for a morphism of pre-$R$-algebroids to be a d-split precrossed module and a necessary and sufficient condition for a d-split$^{\ast}$ precrossed module to be a crossed module. In addition, we examine the hierarchical relations between the categories obtained and look over some results for split (pre)crossed modules over associative $R$-algebras, as a reduced case.