Гомоморфізми матричних груп та кілець над асоціативними кільцями

In the article from the same positions group homomorphisms of matrix groups and ring homomorphisms of matrix rings over associative rings from 1 are described. It is shown that the description of homomorphisms of matrix groups E (n, R) ⊆ G ⊆ GL(n, R), n ≥ 2 into the group of automorphisms GL(W) of the left (optionally free) K-module W over an arbitrary associative ring K of 1 is reduced to cases where 2 or 3 are reversible elements in the ring K. It is proved that they allow a standard description of homomorphisms of the group of elementary transvections $E(n,R), if such a description allows homomorphisms of matrix groups over rings K, in which 2 or 3 are reversible elements. The ring homomorphisms are also described Λ : Rn → End W, n ≥ 2 of the left (optionally free) K-module W over an arbitrary associative ring K of 1. It is shown that homomorphisms Λ allow a standard description on the ring Rn.