Finite rings in varieties with definable principal congruences
Author:
G. E. Simons
Journal:
Proc. Amer. Math. Soc. 121 (1994), 649655
MSC:
Primary 16R10; Secondary 08B26, 16P10, 16R40
MathSciNet review:
1207541
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Abstract 
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Additional Information
Abstract: A variety of rings has definable principal congruences if there is a firstorder formula defining principal twosided ideals for all rings in . Any variety of commutative rings has definable principal congruences, but many noncommutative 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 in a variety with definable principal congruences.
 [AF]
Frank
W. Anderson and Kent
R. Fuller, Rings and categories of modules, 2nd ed., Graduate
Texts in Mathematics, vol. 13, SpringerVerlag, New York, 1992. MR 1245487
(94i:16001)
 [B]
Kirby
A. Baker, Definable normal closures in locally finite varieties of
groups, Houston J. Math. 7 (1981), no. 4,
467–471. MR
658562 (83g:20026)
 [BB]
John
T. Baldwin and Joel
Berman, The number of subdirectly irreducible algebras in a
variety, Algebra Universalis 5 (1975), no. 3,
379–389. MR 0392765
(52 #13578)
 [BL1]
Stanley
Burris and John
Lawrence, Definable principal congruences in varieties of groups
and rings, Algebra Universalis 9 (1979), no. 2,
152–164. MR
523930 (80c:08004), http://dx.doi.org/10.1007/BF02488027
 [BL2]
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
(82j:08008), http://dx.doi.org/10.1007/BF02483839
 [J]
G.
J. Janusz, Separable algebras over commutative
rings, Trans. Amer. Math. Soc. 122 (1966), 461–479. MR 0210699
(35 #1585), http://dx.doi.org/10.1090/S00029947196602106995
 [K]
Emil
W. Kiss, Definable principal congruences in congruence distributive
varieties, Algebra Universalis 21 (1985),
no. 23, 213–224. MR 855740
(88c:08007), http://dx.doi.org/10.1007/BF01188057
 [M1]
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
0469853 (57 #9634)
 [M2]
Ralph
McKenzie, Residually small varieties of 𝐾algebras,
Algebra Universalis 14 (1982), no. 2, 181–196.
MR 634997
(83d:08012b), http://dx.doi.org/10.1007/BF02483919
 [P]
Claudio
Procesi, Rings with polynomial identities, Marcel Dekker Inc.,
New York, 1973. Pure and Applied Mathematics, 17. MR 0366968
(51 #3214)
 [R]
R.
Raghavendran, Finite associative rings, Compositio Math.
21 (1969), 195–229. MR 0246905
(40 #174)
 [Ro]
Louis
H. Rowen, Ring theory, Student edition, Academic Press Inc.,
Boston, MA, 1991. MR 1095047
(94e:16001)
 [S1]
G.
E. Simons, Varieties of rings with definable
principal congruences, Proc. Amer. Math.
Soc. 87 (1983), no. 3, 397–402. MR 684626
(85a:16020), http://dx.doi.org/10.1090/S00029939198306846266
 [S2]
G.
E. Simons, Definable principal congruences and
𝑅stable identities, Proc. Amer. Math.
Soc. 97 (1986), no. 1, 11–15. MR 831376
(87d:16044), http://dx.doi.org/10.1090/S00029939198608313761
 [S3]
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
(92g:16034), http://dx.doi.org/10.1090/S00029947199210531163
 [T]
Sauro
Tulipani, On classes of algebras with the definability of
congruences, Algebra Universalis 14 (1982),
no. 3, 269–279. MR 654395
(83g:08002), http://dx.doi.org/10.1007/BF02483930
 [W]
Robert
S. Wilson, Representations of finite rings, Pacific J. Math.
53 (1974), 647–649. MR 0369423
(51 #5656)
 [AF]
 F. Anderson and K. Fuller, Rings and categories of modules, SpringerVerlag, New York, 1973. MR 1245487 (94i:16001)
 [B]
 K. Baker, Definable normal closures in locally finite varieties of groups, Houston J. Math. 7 (1981), 467471. MR 658562 (83g:20026)
 [BB]
 J. Baldwin and J. Berman, The number of subdirectly irreducible algebras in a variety, Algebra Universalis 5 (1975), 379389. MR 0392765 (52:13578)
 [BL1]
 S. Burris and J. Lawrence, Definable principal congruences in varieties of groups and rings, Algebra Universalis 9 (1979), 152164. MR 523930 (80c:08004)
 [BL2]
 , A correction to ibid., Algebra Universalis 13 (1981), 264267. MR 631561 (82j:08008)
 [J]
 G. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461479. MR 0210699 (35:1585)
 [K]
 E. Kiss, Definable principal congruences in congruence distributive varieties, Algebra Universalis 21 (1985), 213224. MR 855740 (88c:08007)
 [M1]
 R. McKenzie, Paraprimal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties, Algebra Universalis 8 (1978), 336348. MR 0469853 (57:9634)
 [M2]
 , Residually small varieties of Kalgebras, Algebra Universalis 14 (1982), 181196. MR 634997 (83d:08012b)
 [P]
 C. Procesi, Rings with polynomial identities, Marcel Dekker, New York, 1973. MR 0366968 (51:3214)
 [R]
 R. Raghavendran, Finite associative rings, Compositio Math. 21 (1969), 195229. MR 0246905 (40:174)
 [Ro]
 L. H. Rowen, Ring theory, Academic Press, San Diego, 1988. MR 1095047 (94e:16001)
 [S1]
 G. E. Simons, Varieties of rings with definable principal congruences, Proc. Amer. Math. Soc. 87 (1983), 397402. MR 684626 (85a:16020)
 [S2]
 , Definable principal congruences and Rstable identities, Proc. Amer. Math. Soc. 97 (1986), 1115. MR 831376 (87d:16044)
 [S3]
 , The structure of rings in some varieties with definable principal congruences, Trans. Amer. Math. Soc. 331 (1992), 165179. MR 1053116 (92g:16034)
 [T]
 S. Tulipani, On classes of algebras with the definability of congruences, Algebra Universalis 14 (1982), 269279. MR 654395 (83g:08002)
 [W]
 R. S. Wilson, Representations of finite rings, Pacific J. Math. 53 (1974), 643679. MR 0369423 (51:5656)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199412075415
PII:
S 00029939(1994)12075415
Keywords:
Definable principal congruences,
varieties of rings,
finite rings,
subdirectly irreducible rings,
finite local rings
Article copyright:
© Copyright 1994 American Mathematical Society
