PRUFER ☆-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. Prufer, P. Lorenzen and P. Jaffard (cf. Halter-Koch's book [32]), in this paper we outline a general approach to the theory of Prufer star-multiplication domains (or P star MDs), where star 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. Garcia, 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 star MD, when star is a semistar operation, in terms of polynomials (by using the classical characterization of Prufer domains, in terms of polynomials given by R. Gilmer and J. Ho ff man [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 star MD property by ascent and descent in case of field extensions. In this context, we generalize to the P star MD case some classical results concerning Prufer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prufer [51] and W. Krull [41] (cf. also F. Lucius [43] and F. Halter-Koch [34]). Finally, we develop several examples and applications when star 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).