ON SEMISTAR NAGATA RINGS, PRUFER-LIKE DOMAINS AND SEMISTAR GOING-DOWN DOMAINS

Let star be a semistar operation on a domain D. Then the semistar Nagata ring Na(D,star) is a treed domain double left right arrow D is (star) over tilde -treed and the contraction map Spec(Na(D,star)) -> QSpec((star) over tilde)(D) boolean OR {0} is a bijection double left right arrow D is (star) over tilde -treed and (star) over tilde -quasi-Prufer domain. Consequently, if D is a (star) over tilde -Noetherian domain but not a field, then D is (star) over tilde -treed if and only if (star) over tilde -dim(D) = 1. The ring Na(D,star) is a going-down domain if and only if D is (star) over tilde -GD domain and (star) over tilde -quasi-Prufer domain. In general, D is a P star MD double left right arrow Na(D,star) is an integrally closed treed domain double left right arrow 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) congruent to Na (D,star)/P Na(D,star). If D is a P star MD (resp., a (star) over tilde -Noetherian domain; resp., a star-Dedekind domain; resp., a (star) over 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).