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 | References | Similar Articles | Additional Information

Abstract: We study varieties of rings with identity that satisfy an identity of the form , where every term of the polynomial has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let be such a variety. The subdirectly irreducible rings in are shown to be finite local rings and are completely described. This results in structure theorems for the rings in 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 also satisfies an identity of the form . Analogous results are shown to hold for varieties satisfying .

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Additional Information

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