Varieties of rings with definable principal congruences
Author:
G. E. Simons
Journal:
Proc. Amer. Math. Soc. 87 (1983), 397402
MSC:
Primary 16A38; Secondary 08B05, 16A12, 16A70
MathSciNet review:
684626
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A variety of rings has definable principal congruences (DPC) if there is a first order sentence defining principal twosided ideals for all rings in . The key result is that for any ring , does not have DPC if . This allows us to show that if has DPC, then is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring , has DPC if and only if is commutative. An example of a finite, local, noncommutative ring with having DPC is given.
 [1]
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)
 [2]
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)
 [3]
Stanley
Burris, An example concerning definable principal congruences,
Algebra Universalis 7 (1977), no. 3, 403–404.
MR
0441822 (56 #216)
 [4]
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
 [5]
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
 [6]
Stanley
Burris and H.
P. Sankappanavar, A course in universal algebra, Graduate
Texts in Mathematics, vol. 78, SpringerVerlag, New York, 1981. MR 648287
(83k:08001)
 [7]
P.
M. Cohn, Universal algebra, 2nd ed., Mathematics and its
Applications, vol. 6, D. Reidel Publishing Co., Dordrecht, 1981. MR 620952
(82j:08001)
 [8]
K.
E. Eldridge, Orders for finite noncommutative rings with
unity, Amer. Math. Monthly 75 (1968), 512–514.
MR
0230772 (37 #6332)
 [9]
George
Grätzer, Universal algebra, 2nd ed., SpringerVerlag, New
York, 1979. MR
538623 (80g:08001)
 [10]
Nathan
Jacobson, 𝑃𝐼algebras, Lecture Notes in
Mathematics, Vol. 441, SpringerVerlag, Berlin, 1975. An introduction. MR 0369421
(51 #5654)
 [11]
Joachim
Lambek, Lectures on rings and modules, 2nd ed., Chelsea
Publishing Co., New York, 1976. MR 0419493
(54 #7514)
 [12]
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)
 [13]
Louis
Halle Rowen, Polynomial identities in ring theory, Pure and
Applied Mathematics, vol. 84, Academic Press Inc. [Harcourt Brace
Jovanovich Publishers], New York, 1980. MR 576061
(82a:16021)
 [14]
Walter
Taylor, Residually small varieties, Algebra Universalis
2 (1972), 33–53. MR 0314726
(47 #3278)
 [1]
 K. A. Baker, Definable normal closures in locally finite varieties of groups, Houston J. Math. 7 (1981), 467471. MR 658562 (83g:20026)
 [2]
 J. T. Baldwin and J. Berman, The number of subdirectly irreducible algebras in a variety, Algebra Universalis 5 (1979), 379389. MR 0392765 (52:13578)
 [3]
 S. Burris, An example concerning definable principal congruences, Algebra Universalis 7 (1977), 403404. MR 0441822 (56:216)
 [4]
 S. Burris and J. Lawrence, Definable principal congruences in varieties of groups and rings, Algebra Universalis 9 (1979), 152164. MR 523930 (80c:08004)
 [5]
 , A correction to [4], Algebra Universalis 13 (1981), 264267. MR 631561 (82j:08008)
 [6]
 S. Burris and H. P. Sankappanavar, A course in universal algebra, SpringerVerlag, New York, 1981. MR 648287 (83k:08001)
 [7]
 P. M. Cohn, Universal algebra, rev. ed., Reidel, Dordrecht, 1981. MR 620952 (82j:08001)
 [8]
 K. E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly 75 (1968), 512514. MR 0230772 (37:6332)
 [9]
 G. Gratzer, Universal algebra, 2nd ed., SpringerVerlag, New York, 1979. MR 538623 (80g:08001)
 [10]
 N. Jacobson, PIalgebras, Lecture Notes in Math., vol. 441, SpringerVerlag, New York, 1975. MR 0369421 (51:5654)
 [11]
 J. Lambek, Lectures on rings and modules, Chelsea, New York, 1976. MR 0419493 (54:7514)
 [12]
 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)
 [13]
 L. H. Rowen, Polynomial identities in ring theory, Academic Press, New York, 1980. MR 576061 (82a:16021)
 [14]
 W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 3353. MR 0314726 (47:3278)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
16A38,
08B05,
16A12,
16A70
Retrieve articles in all journals
with MSC:
16A38,
08B05,
16A12,
16A70
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198306846266
PII:
S 00029939(1983)06846266
Keywords:
Varieties of rings,
definable principal congruences,
polynomial identity rings
Article copyright:
© Copyright 1983 American Mathematical Society
