The Diophantine problem in the classical matrix groups

In this paper we study the Diophantine problem in the classical matrix groups , , and , , over an associative ring with identity. We show that if is one of these groups, then the Diophantine problem in is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in . When we assume that is commutative. Similar results hold for and provided has no zero divisors (for the ring is not assumed to be commutative).