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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 .

A coefficient ring for finite non-commutative rings