The tensor completion functor in categories of exponential $MR$-groups

The notion of an exponential $R$-group, where $R$ is an arbitrary associative ring with unity, was introduced by R. Lyndon. A. G. Myasnikov and V. N. Remeslennikov refined this notion by adding an extra axiom. In particular, the new notion of an exponential $MR$-group is an immediate generalization of the notion of an $R$-module to the case of noncommutative groups. Basic concepts in the theory of exponential $MR$-groups are presented, and we propose a particular method for constructing tensor completion – the key construction in the category of $MR$-groups. As a consequence, free $MR$-groups and free $MR$-products are described using the language of group constructions.