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

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).

References

D. D. Anderson, J. Matijevic, and W. Nichols, The Krull intersection theorem. II, Pacific J. Math. 66 (1976), no. 1, 15–22. MR 435062
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
John A. Beachy and William D. Weakley, Piecewise Noetherian rings, Comm. Algebra 12 (1984), no. 21-22, 2679–2706. MR 757787, DOI 10.1080/00927878408823127
N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1293, Hermann, Paris, 1961 (French). MR 0171800

Algèbre commutative

Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR 0427289
Robert Gilmer and William Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980), no. 1, 13–16. MR 560575, DOI 10.1090/S0002-9939-1980-0560575-6
William Heinzer and David Lantz, $N$-rings and ACC on colon ideals, J. Pure Appl. Algebra 32 (1984), no. 2, 115–127. MR 741961, DOI 10.1016/0022-4049(84)90047-1
William Heinzer and Jack Ohm, On the Noetherian-like rings of E. G. Evans, Proc. Amer. Math. Soc. 34 (1972), 73–74. MR 294316, DOI 10.1090/S0002-9939-1972-0294316-2
Max. D. Larsen and Paul J. McCarthy, Multiplicative theory of ideals, Pure and Applied Mathematics, Vol. 43, Academic Press, New York-London, 1971. MR 0414528
Chin-Pi Lu, Modules satisfying ACC on a certain type of colons, Pacific J. Math. 131 (1988), no. 2, 303–318. MR 922221
D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968. MR 0231816

Descompunerea primarǎ in inele comutative