Homomorphisms of matrix rings into matrix rings

Let Vn(Rn) be the universal ring with respect to em- beddings of the matrix ring Rn into n X n matrix rings over commutative rings. A construction and a representation is given for this ring. As a main tool in the construction, it is proved that every R homomorphism of Rn9 R a com- mutative ring, is the restriction of an inner automorphism of Un, for some U 2 R. Using this, a necessary and sufficient condition for n2 matrices in Rn to be matrix units is given. 1* Introduction and notationsAll rings to be considered in this paper, except those denoted specifically as matrix rings, will be commutative rings with unit. All homomorphisms are unitary. The unit of a subring coincides with the unit of its over-ring. Denote by Rn the ring of n x n matrices over a ring R. Let ψ R —• S be a ring homomorphism then η induces a homomorphism ηn:Rn-*Sn given by: ηn{ri3) = (^(r^ )). If AeRn,(A)iS will denote the (i, j)th entry of A. The identity element and the standard matrix units of all matrix rings will be denoted by I and {-E^} respectively. Let A be an R algebra. It was proved by Amitsur ((1), Theorem 2) that there exists a commutative R algebra V%(A), and a map p: A— > (V%(A))m which is universal for homomorphisms of A into m x m matrix rings over commutative rings, i.e.; (1) For every τ: A —> Hm9 with H a commutative R algebra, there exists a homomorphism η: VZ(A) —• H such that the following diagram is commutative