Prüfer ⋆--multiplication domains and semistar operations

الأصل · EN

We prove a characterization of a P, 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 Gilmer/Hoffmann:1974, as a model), extending a result proved in the star case by E. Houston, S.J. Malik and J. Mott Houston/Malik/Mott:1984. We also deal with the preservation of the P property by ``ascent'' and ``descent'' in case of field extensions. In this context, we generalize to the P case some classical results concerning Prüfer domains and PvMDs. In particular, we reobtain as a particular case a result due to H. Prüfer Prufer and W. Krull Krull:1936 (cf. also F. Lucius Lucius:1998 and F. Halter-Koch Koch:2000). 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).