Prüfer ⋆-Multiplication Domains and Semistar Operations

Abstract

Starting from the notion of semistar operation, introduced in 1994 by Okabe and Matsuda [49], which generalizes the classical concept of star operation (cf. Gilmer's book [27]) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer, P. Lorenzen and P. Jaffard (cf. Halter-Koch's book [32]), in this paper we outline a general approach to the theory of Prüfer ⋆-multiplication domains (or P⋆MDs), where ⋆ is a semistar operation. This approach leads to relax the classical restriction on the base domain, which is not necessarily integrally closed in the semistar case, and to determine a semistar invariant character for this important class of multiplicative domains (cf. also J.M. García, P. Jara and E. Santos [25]). We give a characterization theorem of these domains in terms of Kronecker function rings and Nagata rings associated naturally to the given semistar operation, generalizing previous results by J. Arnold and J. Brewer [10] and B.G. Kang [39]. We prove a characterization of a P⋆MD, when ⋆ is a semistar operation, in terms of polynomials (by using the classical characterization of Prüfer domains, in terms of polynomials given by . We also deal with the preservation of the P⋆MD property by "ascent" and "descent" in case of field extensions. In this context, we generalize to the P⋆MD case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer [51] and W. Krull [41] (cf. also F. Lucius ). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the "standard" v-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).

Semistar Linkedness and Flatness, Prüfer Semistar Multiplication Domains

“…[21], [9]). In Section 2 we recall the main definitions and we collect some background results on semistar operations.…”

A Note on Prüfer Semistar Multiplication Domains

“…As we mentioned in the introduction, in [5], M. Fontana, P. Jara, and E. Santos have introduced the notion of a P MD to generalize the concepts of Prüfer domains, PvMD's and -multiplication domains to the context of semistar operations.…”

“…After the introduction of semistar operations by Okabe and Matsuda [14], Fontana, Jara and Santos [5] have defined Prüfer semistar multiplication domains (P MD's), as a further generalization to the semistar context of these concepts. To define P MD's they used a notion of -invertibility analogous to the one already used for star operations.…”

A generalization of Kronecker function rings and Nagata rings

“…Since D itself is a * -monolocality (by (8)), then ∧ L is the identity function. However, the proof of (8) above demonstrates that * f (= (∧ L ′ ) f by Proposition 7.3 (4)) is not the identity.…”