Definable principal congruences and 𝑅-stable identities

Simons, G. E.

doi:10.1090/S0002-9939-1986-0831376-1

Definable principal congruences and $R$-stable identities
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by G. E. Simons

Proc. Amer. Math. Soc. 97 (1986), 11-15

DOI: https://doi.org/10.1090/S0002-9939-1986-0831376-1

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Abstract:

We show that an algebra over an infinite field generates a variety with definable principal congruences if and only if it is commutative. A similar result is proved for polynomial rings. The main tool used is the notion from the theory of PI-rings of an $R$-stable identity.

References

Kirby A. Baker, Definable normal closures in locally finite varieties of groups, Houston J. Math. 7 (1981), no. 4, 467–471. MR 658562
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
Nathan Jacobson, $\textrm {PI}$-algebras, Lecture Notes in Mathematics, Vol. 441, Springer-Verlag, Berlin-New York, 1975. An introduction. MR 0369421, DOI 10.1007/BFb0070021
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
Claudio Procesi, Rings with polynomial identities, Pure and Applied Mathematics, vol. 17, Marcel Dekker, Inc., New York, 1973. MR 0366968
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