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Rings with Finite Norm Property

Published online by Cambridge University Press:  20 November 2018

Kathleen B. Levitz  and
Joe L. Mott
Kathleen B. Levitz
Affiliation:
Florida State University, Tallahassee, Florida
Joe L. Mott
Affiliation:
Florida State University, Tallahassee, Florida
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A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].

If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 557 - 565
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Baumschlag, G., Hopficity and abelian groups, in Topics in abelian groups; proceedings of the symposium held June 4-8,1962, edited by Irwin, J. M. and Walker, E. A., pp. 331-336 (Scott, Foresman, Chicago, Illinois, 1963).Google Scholar
2. Butts, H. S. and Smith, W. W., On the integral closure of a domain, J. Sci. Hiroshima Univ. Ser. A-I Math. 30 (1966), 117–122.Google Scholar
3. Chew, K. L. and Lawn, S., Residually finite rings, Can. J. Math. 22 (1970), 92–101.Google Scholar
4. Eakin, P. M., The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278–282.Google Scholar
5. Eakin, P. M. and Heinzer, W., Nonfiniteness infinite dimensional Krull domains, J. Algebra 14 (1970), 333–340.Google Scholar
6. Fuchs, L., Abelian groups, 3rd ed. (Pergamon, London, 1960).Google Scholar
7. Gilmer, R., Integral domains with noetherian subrings, Comment. Math. Helv. 45 (1970), 129–134.Google Scholar
8. Gilmer, R., Multiplicative ideal theory, Vol. I (Queens University Press, Kingston, Ontario, 1968).Google Scholar
9. Goldman, O., On a special class of Dedekind domain, Topology 3 (1964), 113–118.Google Scholar
10. Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, Michigan, 1964).Google Scholar
11. Kaplansky, I., Commutative rings (Allyn and Bacon, Boston, Mass., 1970).Google Scholar
12. Nagata, M., Local rings (Interscience, New York, 1962).Google Scholar