The structure of rings in some varieties with definable principal congruences

Simons, G. E.

doi:10.1090/S0002-9947-1992-1053116-3

The structure of rings in some varieties with definable principal congruences
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by G. E. Simons PDF

Trans. Amer. Math. Soc. 331 (1992), 165-179 Request permission

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 $xy = q(x,y)x$.

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