for commutative rings with identity
Authors:
John Lawrence and Boza Tasic
Journal:
Proc. Amer. Math. Soc. 134 (2006), 943948
MSC (2000):
Primary 06F05, 68Q99
Published electronically:
July 25, 2005
MathSciNet review:
2196024
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.
 1.
George
M. Bergman,
𝐻𝑆𝑃≠𝑆𝐻𝑃𝑆 for
metabelian groups, and related results, Algebra Universalis
26 (1989), no. 3, 267–283. MR 1044848
(91m:08007), http://dx.doi.org/10.1007/BF01211835
 2.
G.
M. Bergman, Partially ordered sets, and minimal systems of
counterexamples, Algebra Universalis 32 (1994),
no. 1, 13–30. MR 1287014
(95j:06001), http://dx.doi.org/10.1007/BF01190814
 3.
Stanley
Burris and H.
P. Sankappanavar, A course in universal algebra, Graduate
Texts in Mathematics, vol. 78, SpringerVerlag, New YorkBerlin, 1981.
MR 648287
(83k:08001)
 4.
S.
Comer and J.
Johnson, The standard semigroup of operators of a variety,
Algebra Universalis 2 (1972), 77–79. MR 0308012
(46 #7127)
 5.
G.
Grätzer and H.
Lakser, The structure of pseudocomplemented
distributive lattices. II. Congruence extension and amalgamation,
Trans. Amer. Math. Soc. 156 (1971), 343–358. MR 0274359
(43 #124), http://dx.doi.org/10.1090/S00029947197102743599
 6.
W.
Nemitz and T.
Whaley, Varieties of implicative semilattices, Pacific J.
Math. 37 (1971), 759–769. MR 0311522
(47 #84)
 7.
Peter
M. Neumann, The inequality of
𝑆𝑄𝑃𝑆 and 𝑄𝑆𝑃 as
operators on classes of groups, Bull. Amer.
Math. Soc. 76
(1970), 1067–1069. MR 0272875
(42 #7756), http://dx.doi.org/10.1090/S000299041970125629
 8.
D.
Pigozzi, On some operations on classes of algebras, Algebra
Universalis 2 (1972), 346–353. MR 0316354
(47 #4901)
 9.
Robert
W. Quackenbush, Structure theory for equational
classes generated by quasiprimal algebras, Trans. Amer. Math. Soc. 187 (1974), 127–145. MR 0327619
(48 #5961), http://dx.doi.org/10.1090/S0002994719740327619X
 10.
Boža
Tasić, On the partially ordered monoid generated by the
operators 𝐻,𝑆,𝑃,𝑃_{𝑠} on classes of
algebras, J. Algebra 245 (2001), no. 1,
1–19. MR
1868180 (2002i:08001), http://dx.doi.org/10.1006/jabr.2001.8914
 11.
Ross
Willard, Three lectures on the RS problem, Algebraic model
theory (Toronto, ON, 1996) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,
vol. 496, Kluwer Acad. Publ., Dordrecht, 1997, pp. 231–254.
MR
1481448 (99d:03029)
 1.
 G. M. Bergman, for metabelian groups, and related results, Algebra Universalis 26 (1989), 267283. MR 1044848 (91m:08007)
 2.
 G. M. Bergman, Partially ordered sets, and minimal systems of counterexamples, Algebra Universalis 32 (1994), 1330. MR 1287014 (95j:06001)
 3.
 S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, SpringerVerlag, New York, 1981. MR 0648287 (83k:08001)
 4.
 S. D. Comer and J. S. Johnson, The standard semigroup of operators of a variety, Algebra Universalis 2 (1972), 7779. MR 308012 (46:7127)
 5.
 G. Grätzer and H. Lakser, The structure of pseudocomplemented distributive lattices. II: Congruence extension and amalgamation, Trans. Amer. Math. Soc. 156 (1971), 343358. MR 0274359 (43:124)
 6.
 W. Nemitz and T. Whaley, Varieties of implicative semilattices, Pacific J. Math. 37 (1971), 759769. MR 0311522 (47:84)
 7.
 P. M. Neumann, The inequality of and as operators on classes of groups, Bull. Amer. Math. Soc. 76 (1970), 10671069. MR 0272875 (42:7756)
 8.
 D. Pigozzi, On some operations on classes of algebras, Algebra Universalis 2 (1972), 346353. MR 0316354 (47:4901)
 9.
 R. W. Quackenbush, Structure theory for equational classes generated by quasiprimal algebras, Trans. Amer. Math. Soc. 187 (1974), 127145. MR 0327619 (48:5961)
 10.
 B. Tasic, On the partially ordered monoid generated by the operators , , , on classes of algebras, Journal of Algebra 245 (2001), 119. MR 1868180 (2002i:08001)
 11.
 R. Willard, Three Lectures on the RS Problem, B. T. Hart et al. (eds.), Algebraic Model Theory, 231254, 1997 Kluwer Academic Publishers. MR 1481448 (99d:03029)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
06F05,
68Q99
Retrieve articles in all journals
with MSC (2000):
06F05,
68Q99
Additional Information
John Lawrence
Affiliation:
University of Waterloo, Department of Pure Mathematics, Waterloo, Ontario, Canada N2L 3G1
Email:
jwlawren@math.uwaterloo.ca
Boza Tasic
Affiliation:
University of Waterloo, Department of Pure Mathematics, Waterloo, Ontario, Canada N2L 3G1
Email:
btasic@math.uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0002993905080664
PII:
S 00029939(05)080664
Keywords:
Class operators,
commutative rings with identity,
partially ordered monoid
Received by editor(s):
November 29, 2001
Received by editor(s) in revised form:
October 28, 2004
Published electronically:
July 25, 2005
Communicated by:
Lance W. Small
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
