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A note on commutative Baer rings

Published online by Cambridge University Press:  09 April 2009

T. P. Speed  and
M. W. Evans
T. P. Speed
Affiliation:
Monash University, Clayton, Victoria
M. W. Evans
Affiliation:
Monash University, Clayton, Victoria
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The class of commutative rings known as Baer rings was first discussed by J. Kist [4], where many interesting properties of these rings were established. Not necessarily commutative Baer rings had previously been studied by I. Kaplansky [3], and by R. Baer himself [1]. In this note we show that commutative Baer rings, which generalize Boolean rings and p-rings, satisfy the Birkhoff conditions for a variety. Next we give a set of equations characterising this variety involving + and * as binary operations, – and as unary operations, and 0 as nullary operation. Finally we describe Baer-subdirectly irreducible commutative Baer rings and state the appropriate representation theorem.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 1 , February 1972 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Baer, R.Linear Algebra and Projective Geometry (Academic Press, 1952).Google Scholar
[2]Cohn, P. M.Universal Algebra (Harper and Row, 1965).Google Scholar
[3]Kaplansky, I.Rings of Operators (mimeographed notes, University of Chicago 1955).Google Scholar
[4]Kist, J. ‘Minimal Prime Ideals in Commutative Semigroups’, Proc. Lond. Math. Soc. (3) 13 (1963) 31–50.CrossRefGoogle Scholar
[5]Speed, T. P., ‘A note on commutative Baer rings’ J. Aust. Math. Soc. (to appear).Google Scholar