Some remarks on Prüfer ⋆-multiplication domains and class groups

Abstract

Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T T T := {T λ | λ ∈ Λ} be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T T T . We show that D is a P * MD (respectively, PvMD) if and only if (c D (f g)) * = (c D (f )c D (g)) * (respectively, (c D (f g)) w = (c D (f )c D (g)) w ) for all 0 = f, g ∈ K[X]. A more general version of this result is given in the semistar operation setting. We give a method for recognizing PvMD's which are not P * MD's for a certain finite type star operation * . We study domains D for which the * -class group Cl * (D) equals the t-class group Cl t (D) for any finite type star operation * , and we indicate examples of PvMD's D such that Cl * (D) Cl t (D). We also compute Cl v (D) for certain valuation domains D.

Ultra Star Operations on Ultra Product of Integral Domains

“…Then it is well known that is a stable star operation on R of finite type called the stable star operation of finite type associated to . It is not hard to see that Max (R)=Max f (R) [4, Corollary 3.5 (2)]. From the latest fact, it then follows that an ideal A is -invertible if and only if it is f -invertible (in fact, if a star operation is of finite type, then (AA −1 ) = R if and only if AA −1 M for all M ∈ Max (R)).…”

Jaffard families and localizations of star operations

“…Therefore, by the inductive hypothesis, Therefore, under each case of Theorem 7.4, the determination of G * (R) is reduced to the calculation of Cl v (V ), where V is a valuation domain. In the case where the maximal ideal M of V is branched (that is, if there is a M-primary ideal of V different from R, or equivalently if there is a prime ideal P M such that there are no prime ideal properly contained between P and M [16, Theorem 17.3]), this group has been calculated in [4,Corollaries 3.6 and 3.7]. Indeed, if P is the prime ideal directly below M, and H is the value group of V /P (represented as a subgroup of R), then…”

“…Suppose now R is finite-dimensional and of finite character, and let Θ be the standard decomposition of R. By Lemma 6.1, there is a bijective correspondence between Θ and the height 1 prime ideals of R, and every T ∈ Θ is semilocal. Hence, by Proposition 6.2 and by the previous case, if there is a prime ideal P M such that there are no prime ideal properly contained between P and M[16, Theorem 17.3]), this group has been calculated in[4, Corollaries 3.6 and 3.7]. Indeed, if P is the prime ideal directly below M, and H is the value group of V /P (represented as a subgroup of R), thenCl v (V ) ≃ 0 if G ≃ Z R/H otherwise.…”