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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The structure of rings in some varieties with definable principal congruences


Author: G. E. Simons
Journal: Trans. Amer. Math. Soc. 331 (1992), 165-179
MSC: Primary 16R10
DOI: https://doi.org/10.1090/S0002-9947-1992-1053116-3
MathSciNet review: 1053116
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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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DOI: https://doi.org/10.1090/S0002-9947-1992-1053116-3
Keywords: Definable principal congruences, varieties of rings, subdirectly irreducible rings, finite local rings, residually small varieties
Article copyright: © Copyright 1992 American Mathematical Society