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Atomic rings and the ascending chain condition for principal ideals

Published online by Cambridge University Press:  24 October 2008

Anne Grams
Affiliation:
University of Tennessee at Nashville, Nashville, Tennessee, U.S.A.

Extract

Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals, or that R has property (M), if each ascending sequence (a1) ⊆ (a2) ⊆ … of principal ideals of R terminates. Property (M) is equivalent to the maximum condition on principal ideals; that is, under the partial order of set containment, each collection of principal ideals of R has a maximum element. Noetherian rings, of course, have property (M), but the converse is not true; for if R has property (M) and if {Xλ} is a set of indeterminates over R, then the polynomial ring R[{Xλ}] has property (M). Krull domains, and hence unique factorization domains, have property (M).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

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