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 $M R$-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 $M R$-groups and of tensor completions of groups in varieties. Abelian varieties of $M R$-groups are described, and various definitions of nilpotency in this category are compared. It turns out that the completion of a $2$-step nilpotent $M R$-group is $2$-step nilpotent.