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