Resumen de On semistar Nagata rings, Prüefer-like domains and semistar going-down domains
David E. Dobbs, Parviz Sahandi
Let star be a semistar operation on a domain D. Then the semistar Nagata ring Na(D, star) is a treed domain if and only if D is star-tilde-treed and the contraction map from Spec(Na(D, star)) to the union of the set of quasi-star-tilde-ideals of D and the singleton set consisting of 0 is a bijection if and only if D is a star-tilde-treed and star-tilde-quasi-Prüefer domain. Consequently, if D is a star-tilde-Noetherian domain but not a field, then D is star- tilde-treed if and only if star-tilde-dim(D)=1. The ring Na(D, star) is a going-down domain if and only if D is a star-tilde-GD domain and a star-tilde-quasi-Prüefer domain. In general, D is a PstarMD if and only if Na(D, star) is an integrally closed treed domain if and only if Na(D, star) is an integrally closed going-down domain. If P is a quasi-star-prime ideal of D, an induced stable semistar operation of finite type, star/P, is defined on D/P. The associated Nagata rings satisfy: Na(D/P, star/P) is isomorphic to Na(D, star)/PNa(D, star). If D is a PstarMD (resp., a star-tilde-Noetherian domain; resp., a star-Dedekind domain; resp., a star-tilde-GD domain), then D/P is a P(star/P)MD (resp., a (star/P)-Noetherian domain; resp., a (star/P)-Dedekind domain; resp., a (star/P)-GD domain).
