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Generalized Discrete Valuation Rings

Published online by Cambridge University Press:  20 November 2018

H.-H. Brungs
H.-H. Brungs*
Affiliation:
The University of Alberta, Edmonton, Alberta
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Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:

(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.

In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).

In the following, R will always denote a ring with unit element satisfying (W).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1404 - 1408
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Baer, R., Kollineationen primàrer Praemoduln (to appear).Google Scholar
2. Baer, R., Dualisierbare Moduln und Praemoduln (to appear).Google Scholar
3. Cohn, P. M., Torsion modules over free ideal rings, Proc. London Math. Soc. (3) 17 (1967), 577–599.Google Scholar
4. Hausdorff, F., Mengenlehre (de Gruyter, Berlin, 1935).Google Scholar
5. Jategaonkar, A. V., A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418–440.Google Scholar
6. Osofsky, B. L., Global dimension of valuation rings, Trans. Amer. Math. Soc. 127 (1967), 136–149.Google Scholar