2002/11/26 by Marco Fontana, Fontana, Marco, Pascual Jara +3
Mathematics · #13A15 #13A18 #13F05 #13F30 #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #msc:13A15 #msc:13A18 #msc:13F05 #msc:13F30
paper · pdf · doi:10.48550/arxiv.math/0211410
arxiv created 2002/11/26 · arxiv updated 2009/11/30
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 \citeGilmer/Hoffmann:1974, as a model), extending a result proved in the star case by E. Houston, S.J. Malik and J. Mott \citeHouston/Malik/Mott:1984. 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 \citePrufer and W. Krull \citeKrull:1936 (cf. also F. Lucius \citeLucius:1998 and F. Halter-Koch \citeKoch: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).