Varieties of Exponential R-Groups

The notion of an exponential R -group, where R is an arbitrary associative ring with unity, was introduced by R. Lyndon. Myasnikov and Remeslennikov refined the notion of an R -group by introducing an additional axiom. In particular, the new concept of an exponential MR -group ( R -ring) is a direct generalization of the concept of an R -module to the case of noncommutative groups. We come up with the notions of a variety of MR -groups and of tensor completions of groups in varieties. Abelian varieties of MR -groups are described, and various definitions of nilpotency in this category are compared. It turns out that the completion of a 2-step nilpotent MR -group is 2-step nilpotent.