Pr\"{u}fer star--multiplication domains and semistar operations

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 Pr\"{u}fer domains, in terms of polynomials given by R. Gilmer and J. Hoffman \cite{Gilmer/Hoffmann:1974}, as a model), extending a result proved in the star case by E. Houston, S.J. Malik and J. Mott \cite{Houston/Malik/Mott:1984}. 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 Pr\"{u}fer domains and P$v$MDs. In particular, we reobtain as a particular case a result due to H. Pr\"{u}fer \cite{Prufer} and W. Krull \cite{Krull:1936} (cf. also F. Lucius \cite{Lucius:1998} and F. Halter-Koch \cite{Koch:2000}). 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).