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Elementary Factorization in π-Regular Rings

Published online by Cambridge University Press:  20 November 2018

Arthur Steger
Arthur Steger*
Affiliation:
University of New Mexico, Albuquerque, New Mexico
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This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966, pp. 307 - 313
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Foster, A. L., The theory of Boolean-like rings, Trans. Amer. Math. Soc., 59 (1946), 166–187.Google Scholar
2. Foster, A. L., The idempotent elements of a commutative ring form a Boolean algebra, Duke Math.J., 12 (1945), 143–152.Google Scholar
3. McCoy, N. H., Generalized regular rings, Bull. Amer. Math. Soc, 45 (1939), 175–178.Google Scholar
4. Pollak, G., Ueber die Struktur kommutativer Hauptidealringe, Acta Scient. Math., 22 (1961), 62–74.Google Scholar
5. van der Waerden, B. I., Moderne Algebra, 2nd éd., vol. I (New York, 1940), p. 63.Google Scholar