On the cardinality of semistar operations of finite character on integral domains
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let D be an integral domain with Spec(D) finite, K the quotient field of D, [D,K] the set of rings between D and K, and SFc(D) the set of semistar operations of finite character on D. It is well known that |Spec(D)| leq |SFc(D)|. In this paper, we prove that |Spec(D)| = |SFc(D)| if and only if D is a valuation domain, if and only if |Spec(D)| = |[D,K]|. We also study integral domains D such that |Spec(D)| +1 = |SFc(D)|.
Abstract
Let D be an integral domain with Spec(D) finite, K the quotient field of D, [D,K] the set of rings between D and K, and SFc(D) the set of semistar operations of finite character on D. It is well known that |Spec(D)| leq |SFc(D)|. In this paper, we prove that |Spec(D)| = |SFc(D)| if and only if D is a valuation domain, if and only if |Spec(D)| = |[D,K]|. We also study integral domains D such that |Spec(D)| +1 = |SFc(D)|.
참고문헌 (11)
- M. Fontana, "Localizing systems and semistar operations, Non-Noetherian Commutative Ring Theory, Vol. 520", pp.169~197 (2000), Kluwer Academic Publisher
- M. Fontana, "Nagata rings, Kronecker function rings and related semistar operations", Comm. Algebra, 31, pp.4775~4805 (2003)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Marcel Dekker
- E. Houston, "On t-invertibility II", Comm. Algebra, 17, pp.1955~1969 (1989)
- Nv, "J. Algebra", 1989, 151~170
- R. Matsuda, "Note on valuation rings and semistar operations", Comm. Algebra, 28, pp.2515~1519 (2000)
- R. Matsuda, "Commutative Rings", pp.77~81 (2002), Nova Science Publisher
- A. Mimouni, "Semistar-operations of finite character on integral domains", J. Pure Appl. Algebra, 200, pp.37~50 (2005)
- A. Mimouni, "Semistar operations on valuation domains", Internat. J. Commutative Rings, 2, pp.131~141 (2003)
- A. Okabe, "Semistar operations on integral domains", Math. J. Toyama Univ, 17, pp.1~21 (1994)
- G. Picozza, "Star operations on overrings and semistar operations", Comm. Algebra, 33, pp.2051~2073 (2005)