Abstract:
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.
