Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

Abstract

Abstract. In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book [20]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's.In [17] and [18] the current authors investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D. In this paper we extend that study and also generalize Kang's notion of a star Nagata ring [30] and [31] to the semistar setting. Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

“…This concept has been proven, regarding its flexibility, extremely useful in studying the structure of different classes of integral domains (cf. for instance [28], [8], [10], [11], [12], and [20]). …”

Semistar-Krull and valuative dimension of integral domains

“…Using [17,Corollary 19.7], there exists a valuation overring V of T Pn , such that V contains a chain M 0 ⊂ M 1 ⊂ · · · ⊂ M n of prime ideals of V and M i ∩ T Pn = P i T Pn . Since P n ∈ QMax e ⋆ι (T ) and V is an overring of T Pn , we obtain that V is a ⋆ ι -valuation overring of T , by [14,Theorem 3.9]. So that V e ⋆ι = V , (see [10,Page 34]).…”

“…Let V be a valuation overring of T . Since D P ⊆ V , and P is a quasi-⋆-maximal ideal of D, we have V is a ⋆-valuation overring of D by [14,Theorem 3.9]. Thus V e ⋆ = V by [10,Page 34].…”

Ring-Theoretic Properties of PvMDs

“…If in addition T is t-linked over R, the w-operation is a star operation on T [8, Proposition 3.16]. Note that this star operation, being spectral and of finite type [12], is generally smaller than the w-operation on T , that we denote by w ′ to avoid confusion. Proposition 1.5.…”