Some results of the theory of exponential $R$-groups

This paper is devoted to the study of groups from the category $\frak{M}$ of $R$-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of $A$-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent $R$-groups of nilpotency class 2, where $R$ is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [2]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial $R$-power groups which are embeddable to their $A$-tensor completions. The free $R$-groups and free $R$-products are described with usual group-theoretical free constructions.