A coefficient ring for finite non-commutative rings

Abstract

We prove that every finite p-ring R contains a unique (up to isomorphism) subring S such that S / p S ≅ R / rad R S/pS \cong R/{\operatorname {rad}}\;R . S is shown to be a direct sum of full matrix rings over rings of the form Z p n [ x ] / ( f ( x ) ) {Z_{{p^n}}}[x]/(f(x)) where f ( x ) f(x) is monic and irreducible modulo p.