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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Regularity conditions in nonnoetherian rings


Author: T. Kabele
Journal: Trans. Amer. Math. Soc. 155 (1971), 363-374
MSC: Primary 13.95
DOI: https://doi.org/10.1090/S0002-9947-1971-0274439-8
MathSciNet review: 0274439
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Abstract: We show that properties of $ R$-sequences and the Koszul complex which hold for noetherian local rings do not hold for nonnoetherian local rings. For example, we construct a local ring with finitely generated maximal ideal such that $ {\text{hd} _R}M < \infty $ but $ M$ is not generated by an $ R$-sequence. In fact, every element of $ M - {M^2}$ is a zero divisor. Generalizing a result of Dieudonné, we show that even in local (nonnoetherian) integral domains a permutation of an $ R$-sequence is not necessarily an $ R$-sequence.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0274439-8
Keywords: Nonnoetherian local ring, regular sequence, quasi-regular sequence, Koszul complex
Article copyright: © Copyright 1971 American Mathematical Society