Finite rings in varieties with definable principal congruences
HTML articles powered by AMS MathViewer
- by G. E. Simons
- Proc. Amer. Math. Soc. 121 (1994), 649-655
- DOI: https://doi.org/10.1090/S0002-9939-1994-1207541-5
- PDF | Request permission
Abstract:
A variety $\mathcal {V}$ of rings has definable principal congruences if there is a first-order formula defining principal two-sided ideals for all rings in $\mathcal {V}$. Any variety of commutative rings has definable principal congruences, but many non-commutative rings cannot be in a variety with definable principal congruences. We show that a finite ring in a variety with definable principal congruences is a direct product of finite local rings. This result is used to describe the structure of all finite rings R with $J{(R)^2} = 0$ in a variety with definable principal congruences.References
- Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
- 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 H. Rowen, Ring theory, Student edition, Academic Press, Inc., Boston, MA, 1991. MR 1095047
- 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
- G. E. Simons, The structure of rings in some varieties with definable principal congruences, Trans. Amer. Math. Soc. 331 (1992), no. 1, 165–179. MR 1053116, DOI 10.1090/S0002-9947-1992-1053116-3
- 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
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 649-655
- MSC: Primary 16R10; Secondary 08B26, 16P10, 16R40
- DOI: https://doi.org/10.1090/S0002-9939-1994-1207541-5
- MathSciNet review: 1207541