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Rings with No Nilpotent Elements and with the Maximum Condition on Annihilators

Published online by Cambridge University Press:  20 November 2018

W. H. Cornish  and
P. N. Stewart
W. H. Cornish
Affiliation:
The Flinders University of South Australia, Bedford Park, South Australia
P. N. Stewart
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Rings (all of which are assumed to be associative) with no non-zero nilpotent elements will be called reduced rings; R is a reduced ring if and only if x2=0 implies x=0, for all x∈R. In 2. we prove that the following conditions on an annihilator ideal I of a reduced ring are equivalent: I is a maximal annihilator, I is prime, I is a minimal prime, I is completely prime. A characterization of reduced rings with the maximum condition on annihilators is given in 3.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 35 - 38
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Divinsky, N. J., Rings and radicals, University of Toronto Press, Toronto, 1965.Google Scholar
2. Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, 1966.Google Scholar