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Divisibility Properties of Graded Domains

Published online by Cambridge University Press:  20 November 2018

D. D. Anderson  and
David F. Anderson
D. D. Anderson
Affiliation:
The University of Iowa, Iowa City, Iowa
David F. Anderson
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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Let R = ⊕α∊гRα be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z+ or Z-graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 1 , 01 February 1982 , pp. 196 - 215
Copyright
Copyright © Canadian Mathematical Society 1982

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