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Subdirectly Irreducible Semirings and Semigroups Without Nonzero Nilpotents

Published online by Cambridge University Press:  20 November 2018

William H. Cornish
William H. Cornish*
Affiliation:
The Flinders University of South Australia, Bedford Park, Australia
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It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 1 , March 1973 , pp. 45 - 47
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bell, H. E., Duo rings: some applications to commutativity theorems, Canad. Math. Bull. 11 (1968), 375–380.Google Scholar
2. Birkhoff, G., Lattice theory, Colloq. Publ. XXV, 3rd ed., Amer. Math. Soc, Providence, R.I., 1967.Google Scholar