The structure of rings in some varieties with definable principal congruences

Abstract

We study varieties of rings with identity that satisfy an identity of the form x y = y p ( x , y ) xy = yp(x,y) , where every term of the polynomial p p has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let V \mathcal {V} be such a variety. The subdirectly irreducible rings in V \mathcal {V} are shown to be finite local rings and are completely described. This results in structure theorems for the rings in V \mathcal {V} and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that V \mathcal {V} also satisfies an identity of the form x y = q ( x , y ) x xy = q(x,y)x . Analogous results are shown to hold for varieties satisfying x y = q ( x , y ) x xy = q(x,y)x .