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On rings all of whose factor rings are integral domains

Published online by Cambridge University Press:  09 April 2009

Yasuyuki Hirano
Yasuyuki Hirano
Affiliation:
Okayama University, Okayama, 700, Japan
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Abstract

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A ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 3 , December 1993 , pp. 325 - 333
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Feigeistock, S., Additive groups of rings, vol. II, Research Notes in Mathematics 169 (Pitman, Boston, 1988).Google Scholar
[2]Fuchs, L., Infinite abelian groups, vol. I (Academic Press, New York, 1970).Google Scholar
[3]Herstein, I. N., Rings with involution (University of Chicago Press, Chicago, 1976).Google Scholar
[4]Jategaonkar, V. A., ‘A multiplicative analog of the Weyl algebra’, Comm. Algebra 12 (1984), 1669–1688.CrossRefGoogle Scholar
[5]Renault, G., Algébre non commutative (Gauthier-Villars, Paris, 1975).Google Scholar
[6]Rowen, L. H., Polynomial identities in ring theory (Academic Press, New York, 1980).Google Scholar