Definable principal congruences in varieties of groups and rings

Abstract

In [lJ Baldwin and Berman showed that for varieties ~ with DPC (definable principal congruences) certain results of Taylor concerning residually small varieties could be sharpened. Their question as to whether every variety generated by a finite algebra has DPC was answered in the negative in [2]; however the question remained open for varieties with permutable congruences. The study of DPC became even more interesting when McKenzie [4] proved that this property could be used, in certain cases (such as a variety generated by a para-primal algebra), to give an easy proof of the finite axiomatizablity of the variety. McKenzie then showed that among lattices only the distributive varieties have DPC, and states that the question of whether varieties generated by a finite group or ring have DPC is open. In the first section we point out that a variety ~ has DPC iff the free algebra on countably many generators in ~ has SDPC (strongly definable principal congruences), hence a variety generated by a class Y{ of algebras has DPC iff the quasi-variety generated by 5g has DPC. In the second section a finite ring R is constructed such that the variety generated by R does not have DPC. In the third section we prove that if the variety generated by a finite group G has DPC then G must be nilpotent; on the other hand if G is nilpotent class 2 and finite then indeed it generates a variety with DPC. It follows that the properties of having DPC and being finitely axiomatizable are independent for quasi-varieties generated by a finite group. Finally Baldwin's theorem (3) that the variety of all groups of exponent 3 has DPC is shown to be best possible for Burnside varieties.