Integral domains with finitely many spectral semistar operations

Abstract

Let D be a finite-dimensional integral domain, Spec(D) be the set of prime ideals of D, and SpSS(D) be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with |SpSS(D)| = n + dim(D) for 1≤n≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that |SpSS(D)| = n+dim(D) for all positive integers n with n ≠2. As corollaries, we completely characterize the quasi-local domains D with |SpSS(D)| = n+dim(D) for n = 4, 5. Furthermore, we also present the lower and upper bounds of |SpSS(D)| when Spec(D) is a finite tree.

Keywords

(Spectral) semistar operation / prime spectrum / (Krull) dimension

References

[1] Chang G W. On the cardinality of stable star operations of finite type on an integral domain. C R Math Acad Sci Paris, 2012, 350: 557–560

[2] Finocchiaro C A, Fontana M, Spirito D. New distinguished classes of spectral spaces: a survey. arXiv: 1510.04443

[3] Fontana M, Huckaba J. Localizing systems and semistar operations. In: Non-Noetherian Commutative Ring Theory. Math Appl, Vol 520. Dordredt: Kluwer Acad, 2000, 169–197

[4] Fontana M, Loper K. Nagata rings, Kronecker function rings, and related semistar operations. Comm Algebra, 2003, 31: 4775–4805

[5] Gilmer R. Multiplicative Ideal Theory. Queen’s Papers in Pure and Appl Math, Vol 90. Queen’s University, Kingston, Ontario, Canada, 1992

[6] Lewis W J. The spectrum of a ring as a partially ordered set. J Algebra, 1973: 25: 419–434

[7] Mimouni A. Semistar-operations of finite character on integral domains. J Pure Appl Algebra, 2005, 200: 37–50

[8] Mimouni A. On the prime spectrum of an integral domains with finite spectral semistar operations. Algebra Colloq, 2011, 18: 965–972

[9] Okabe A, Matsuda R. Semistar operations on integral domains. Math J Toyama Univ, 1994, 17: 1–21

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg