Finite rings in varieties with definable principal congruences

Abstract

A variety V \mathcal {V} of rings has definable principal congruences if there is a first-order formula defining principal two-sided ideals for all rings in V \mathcal {V} . Any variety of commutative rings has definable principal congruences, but many non-commutative rings cannot be in a variety with definable principal congruences. We show that a finite ring in a variety with definable principal congruences is a direct product of finite local rings. This result is used to describe the structure of all finite rings R with J ( R ) 2 = 0 J{(R)^2} = 0 in a variety with definable principal congruences.