Projections of Finite Rings

Abstract

Let R and R^φ be associative rings with isomorphic subring lattices, and φ be a lattice isomorphism (or else a projection) of the ring R onto the ring R^φ. We call R^φ the projective image of a ring R and call R itself the projective preimage of a ring R^φ. The main result of the first part of the paper is Theorem 5, which proves that the projective image R^φ of a one-generated finite p-ring R is also one-generated if R^φ at the same time is itself a p-ring. In the second part, we continue studying projections of matrix rings. The main result of this part is Theorems 6 and 7, which prove that if R = M_n(K) is the ring of all square matrices of order n over a finite ring K with identity, and φ is a projection of the ring R onto the ring R^φ, then R^φ = M_n(K′), where K′ is a ring with identity, lattice-isomorphic to the ring K.