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$HSP\not = SHPS$ for commutative rings with identity


Authors: John Lawrence and Boza Tasic
Journal: Proc. Amer. Math. Soc. 134 (2006), 943-948
MSC (2000): Primary 06F05, 68Q99
DOI: https://doi.org/10.1090/S0002-9939-05-08066-4
Published electronically: July 25, 2005
MathSciNet review: 2196024
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Abstract: Let $I$, $H$, $S$, $P$, $P_s$ be the usual operators on classes of rings: $I$ and $H$ for isomorphic and homomorphic images of rings and $S$, $P$, $P_s$ respectively for subrings, direct, and subdirect products of rings. If $\mathcal K$ is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class $HSP({\mathcal K})$ is known to be the variety generated by the class $\mathcal K$. Although the class $SHPS({\mathcal K})$ is in general a proper subclass of the class $HSP({\mathcal K})$ for many familiar varieties $HSP({\mathcal K})= SHPS({\mathcal K})$. Our goal is to give an example of a class $\mathcal K$ of commutative rings with identity such that $HSP({\mathcal K})\not = SHPS({\mathcal K})$. As a consequence we will describe the structure of two partially ordered monoids of operators.


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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: https://doi.org/10.1090/S0002-9939-05-08066-4
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.

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