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Atomic rings and the ascending chain condition for principal ideals

  • Anne Grams (a1)
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).

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COPYRIGHT: © Cambridge Philosophical Society 1974
References
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(1)Arnold, J. T.On the ideal theory of the Kronecker function ring and the domain D(X). Canad. J. Math. 21 (1968), 558–563.
(2)Arnold, J. T. and Gilmer, R.Idempotent ideals and unions of nets of Prüfer domains, J. Sci. Hiroshima Univ. Ser. A.-1. Math. 31 (1967), 131–145.
(3)Claborn, L.Specified relations in the ideal group. Michigan Math. J. 15 (1968), 249–255.
(4)Cohen, I. S. and Zariski, O.A fundamental inequality in the theory of extensions of valuations. Illinois J. Math. 1 (1957), 1–8.
(5)Cohn, P. M.Bezout rings and their subrings. Proc. Cambridge Philos. Soc. 64 (1968), 251–264.
(6)Gilmer, R.Multiplicative Ideal Theory (Kingston, Ontario, 1968).
(7)Gilmer, R. and Heinzer, W.Overrings of Prüfer domains. II. J. Algebra 7 (1967), 281–302.
(8)Heinzer, W. and Ohm, J.Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158 (1971), 273–284.
(9)Jaffard, P.Théorie arithmétique des anneaux du type de Dedekind. Bull. Soc. Math. France 80 (1952), 61–100.
(10)Krull, W.Aligemeine Bewertungstheorie. J. Reine Angew. Math. 167 (1931), 160–196.
(11)Mott, J. L. Groups of divisibility. (Preprint.)
(12)Nagata, M.A remark on the unique factorization theorem. J. Math. Soc. Japan 9 (1957), 143–145.
(13)Nakano, N.Idealtheorie in einem speziellen unendlichen algebraischen Zählkorper. J. Sci. Hiroshima Univ. Ser. A. 16 (1953), 425–439.
(14)Ohm, J.Some counterexamples related to integral closure in D[[X]]. Trans. Amer. Math. Soc. 122 (1966), 321–333.
(15)Ohm, J.Semi-valuations and groups of divisibility. Canad. J. Math. 21 (1969), 576–591.
(16)Roquette, P.On the prolongation of valuations. Trans. Amer. Math. Soc. 88 (1958), 42–56.
(17)Samuel, P.Sur les anneaux factoriels. Bull. Soc. Math. France 89 (1961), 155–173.
(18)Samuel, P.Unique factorization. Amer. Math. Monthly 75 (1968), 945–952.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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