GOING-DOWN AND SEMISTAR OPERATIONS

If D ⊆ T is an extension of (commutative integral) domains and ⋆ (resp., ⋆′) is a semistar operation on D (resp., T), we define what it means for D ⊆ T to satisfy the (⋆,⋆′)-GD property. Sufficient conditions are given for (⋆,⋆′)-GD, generalizing classical sufficient conditions for GD such as flatness, openness of the contraction map of spectra and the hypotheses of the classical going-down theorem. If ⋆ is a semistar operation on a domain D, we define what it means for D to be a ⋆-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a [Formula: see text] domain, the domain extensions T of D for which [Formula: see text] is tested can be the [Formula: see text]-valuation overrings of D, the simple overrings of D, or all T. P ⋆ MD s are characterized as the [Formula: see text]-treed (resp., [Formula: see text]) domains D which are [Formula: see text]-finite conductor domains such that [Formula: see text] is integrally closed. Several characterizations are given of the [Formula: see text]-Noetherian domains D of [Formula: see text]-dimension 1 in terms of the behavior of the (⋆,⋆′)-linked overrings of D and the ⋆-Nagata rings Na(D,⋆).

PRÜFER ⋆-MULTIPLICATION DOMAINS AND SEMISTAR OPERATIONS

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 R. Gilmer and J. Hoffman [28], as a model), extending a result proved in the star case by E. Houston, S. J. Malik and J. Mott [36]. 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 [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when ⋆ is a (semi)star given explicitly (e.g. we consider the case of the standardv-, t-, b-, w-operations or the case of semistar operations associated to appropriate families of overrings).