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Modules and rings satisfying (accr)


Author: Chin-Pi Lu
Journal: Proc. Amer. Math. Soc. 117 (1993), 5-10
MSC: Primary 13E05; Secondary 13C05, 13F05, 13J10
DOI: https://doi.org/10.1090/S0002-9939-1993-1104398-7
MathSciNet review: 1104398
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Abstract: A module $ M$ over a ring $ R$ is said to satisfy (accr) if the ascending chain of residuals of the form $ N: B \subseteq N:{B^2} \subseteq N:{B^3} \subseteq \cdots $ terminates for every submodule $ N$ and every finitely generated ideal $ B$ of $ R$. A ring satisfies (accr) if it does as a module over itself. This class of rings and modules satisfies various properties of Noetherian rings and modules. For each of the following rings, we investigate a necessary and sufficient condition for the ring to satisfy (accr): polynomial rings, power series rings, valuation rings, and Prüfer domains. We also prove that if $ R$ is a ring satisfying (accr), then every finitely generated $ R$-module satisfies (accr).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1104398-7
Keywords: (accr), Laskerian ring, valuation ring, Prüfer domain
Article copyright: © Copyright 1993 American Mathematical Society

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