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

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

DOI: https://doi.org/10.1090/S0002-9947-1992-1053116-3

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

References

Kirby A. Baker, Definable normal closures in locally finite varieties of groups, Houston J. Math. 7 (1981), no. 4, 467–471. MR 658562
John T. Baldwin and Joel Berman, The number of subdirectly irreducible algebras in a variety, Algebra Universalis 5 (1975), no. 3, 379–389. MR 392765, DOI 10.1007/BF02485271
Stanley Burris and John Lawrence, Definable principal congruences in varieties of groups and rings, Algebra Universalis 9 (1979), no. 2, 152–164. MR 523930, DOI 10.1007/BF02488027
S. Burris and J. Lawrence, A correction to: “Definable principal congruences in varieties of groups and rings” [Algebra Universalis 9 (1979), no. 2, 152–164; MR 80c:08004], Algebra Universalis 13 (1981), no. 2, 264–267. MR 631561, DOI 10.1007/BF02483839
G. J. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461–479. MR 210699, DOI 10.1090/S0002-9947-1966-0210699-5
Emil W. Kiss, Definable principal congruences in congruence distributive varieties, Algebra Universalis 21 (1985), no. 2-3, 213–224. MR 855740, DOI 10.1007/BF01188057
Ralph McKenzie, Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties, Algebra Universalis 8 (1978), no. 3, 336–348. MR 469853, DOI 10.1007/BF02485404
Ralph Freese and Ralph McKenzie, Residually small varieties with modular congruence lattices, Trans. Amer. Math. Soc. 264 (1981), no. 2, 419–430. MR 603772, DOI 10.1090/S0002-9947-1981-0603772-9
Claudio Procesi, Rings with polynomial identities, Pure and Applied Mathematics, vol. 17, Marcel Dekker, Inc., New York, 1973. MR 0366968
R. Raghavendran, Finite associative rings, Compositio Math. 21 (1969), 195–229. MR 246905
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
G. E. Simons, Varieties of rings with definable principal congruences, Proc. Amer. Math. Soc. 87 (1983), no. 3, 397–402. MR 684626, DOI 10.1090/S0002-9939-1983-0684626-6
G. E. Simons, Definable principal congruences and $R$-stable identities, Proc. Amer. Math. Soc. 97 (1986), no. 1, 11–15. MR 831376, DOI 10.1090/S0002-9939-1986-0831376-1
Sauro Tulipani, On classes of algebras with the definability of congruences, Algebra Universalis 14 (1982), no. 3, 269–279. MR 654395, DOI 10.1007/BF02483930
Robert S. Wilson, Representations of finite rings, Pacific J. Math. 53 (1974), 647–649. MR 369423, DOI 10.2140/pjm.1974.53.643