ON v–DOMAINS AND STAR OPERATIONS

Let ∗ be a star operation on an integral domain D. Let f(D) be the set of all nonzero finitely generated fractional ideals of D. Call D a ∗–Prüfer (respectively, (∗, v)–Prüfer) domain if (FF−1)∗ = D (respectively, (F vF−1)∗ = D) for all F ∈ f(D). We establish that ∗–Prüfer domains (and (∗, v)–Prüfer domains) for various star operations ∗ span a major portion of the known generalizations of Prüfer domains inside the class of v–domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York–London, 1971], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for ∗–Prüfer domains, and which cannot be. We also show that in a ∗–Prüfer domain, each pair of ∗-invertible ∗-ideals admits a GCD in the set of ∗-invertible ∗-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v–multiplication domains. We also link D being ∗–Prüfer (or (∗, v)–Prüfer) with the group Inv∗(D) of ∗-invertible ∗-ideals (under ∗-multiplication) being