Eakin-Nagata theorem for commutative rings whose regular ideals are finitely generated
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (8)
- E. Bastida, "Overrings and divisorial ideals of rings of the form D +M", Michigan Math. J., 20, pp.79~94 (1973)
- G. W. Chang, "Integral closure of a Marot ring whose regular ideals are finitely generated", Comm. Algebra, 27, pp.1783~1795 (1999)
- G. W. Chang, "On Krull overrings of a Marot ring whose regular ideals are finitely generated", Comm. Algebra, 28, pp.2533~2542 (2000)
- P. M. Eakin, "The converse to a well known theorem on Noetherian rings", Math. Anna., 177, pp.278~282 (1968)
- J. A. Huckaba, "Commutative Rings with Zero Divisors" (1988), Dekker
- I. Kaplansky, "Commutative Rings" (1974), The Univ. Chicago Press
- M. Nagata, "A type of subrings of a Noetherian ring", J. Math. Kyoto Univ., 8, pp.465~467 (1968)
- O. Zariski, "Commutative Algebra, Vol. 1" (1958), Princepton