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A note on subdirectly irreducible rings

Published online by Cambridge University Press:  17 April 2009

Shalom Feigelstock
Shalom Feigelstock
Affiliation:
Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, Israel.
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Abstract

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Let R be a commutative subdirectly irreducible ring, with minimal ideal M. It is shown that either R is a field, or M2 = 0. A construction is given which yields commutative sub-directly irreducible rings possessing nonzero-divisors, and nonzero nilpotent elements either with a unity element, or without. Such a ring without unity has been constructed by Divinsky. The same technique enables the construction of subdirectly irreducible rings with mixed additive groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 1 , August 1984 , pp. 137 - 141
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Divinsky, N., “Commutative subdirectly irreducible rings”, Proc. Amer. Math. Soc. 8 (1957), 642–648.CrossRefGoogle Scholar
[2]Feigelstock, S., “The additive groups of subdirectly irreducible rings”. Bull. Austral. Math. Soc. 20 (1979), 164–170.CrossRefGoogle Scholar
[3]Feigelstock, S., Additive Groups of Rings (Pitman, Boston-London, 1983).Google Scholar
[4]Fuchs, L., Infinite Abelian Groups I (Academic Press, New York-London, 1971).Google Scholar
[5]Kruse, R. L. and Price, D. T., Nilpotent Rings (Gordon and Breach, New York-London, 1969).Google Scholar
[6]McCoy, N., “Subdirectly irreducible commutative rings”. Duke Math. J. 12 (1945), 381–387.CrossRefGoogle Scholar