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Let R be a commutative ring with identity, T(R) be the total quotient ring of R, and D be a ring such that R ⊆ D ⊆ T(R) and D is a finite R-module. In this paper, we show that each regular ideal of R is finitely generated if and only if each regular ideal of D is finitely generated. This is a generalization of the Eakin-Nagata theorem that R is Noetherian if and only if D is Noetherian.
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참고문헌 (8)
참고문헌 신청
E. Bastida / 1973 / Overrings and divisorial ideals of rings of the form D +M / Michigan Math. J. 20 : 79 ~ 94
G. W. Chang / 1999 / Integral closure of a Marot ring whose regular ideals are finitely generated / Comm. Algebra 27 : 1783 ~ 1795
G. W. Chang / 2000 / On Krull overrings of a Marot ring whose regular ideals are finitely generated / Comm. Algebra 28 : 2533 ~ 2542
I. Kaplansky / 1974 / Commutative Rings / The Univ. Chicago Press
J. A. Huckaba / 1988 / Commutative Rings with Zero Divisors / Dekker
G. W. Chang / 1999 / Integral closure of a Marot ring whose regular ideals are finitely generated / Comm. Algebra 27 : 1783 ~ 1795
G. W. Chang / 2000 / On Krull overrings of a Marot ring whose regular ideals are finitely generated / Comm. Algebra 28 : 2533 ~ 2542
I. Kaplansky / 1974 / Commutative Rings / The Univ. Chicago Press
J. A. Huckaba / 1988 / Commutative Rings with Zero Divisors / Dekker
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