Modules and rings satisfying (accr)

Lu, Chin-Pi

doi:10.1090/S0002-9939-1993-1104398-7

Modules and rings satisfying (accr)
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by Chin-Pi Lu

Proc. Amer. Math. Soc. 117 (1993), 5-10

DOI: https://doi.org/10.1090/S0002-9939-1993-1104398-7

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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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Descompunerea primarǎ in inele comutative