Isomorphisms of general linear groups over associative rings graded by a commutative group

Isomorphisms of general linear groups over associative rings graded by a commutative is studied by a general theorem. The description of isomorphisms of the groups of invertible elements of endomorphism rings over free graded modules over associative rings graded by a commutative group is found. Two associative rings, R and S and group isomorphism are considered. It is found that there exist central idempotents e and f of the two rings, a ring isomorphism and a ring antiisomorphism. A ring is called G-graded if it is a system of additive subgroups of the ring R. For a commutative group with associative graded rings are graded matrix rings, there exists a group isomorphism.