On computable subgroups of the group of all unitriangular matrices over a ring

The problems of existence and uniqueness of computable numberings are fundamental in theory of computably numbered groups. In connection with the development of the theory of algorithms a study of the problems of computability of important classes of algebraic systems are currently relevant. Groups of unitriangular matrices over the ring are a classic representative of the class of nilpotent groups and have numerous applications both in group theory and in its applications. In this paper we obtain a criterion of computability of subgroups of the group of all unitriangular matrices UTn(K) over a computable associative ring with unity.