Uppers to zero and semistar operations in polynomial rings

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type [⋆] on the polynomial ring D[X], such that D is a ⋆-quasi-Prüfer domain if and only if each upper to zero in D[X] is a quasi-[⋆]-maximal ideal. This result completes the investigation initiated by Houston–Malik–Mott [E. Houston, S. Malik, J. Mott, Characterizations of ∗-multiplication domains, Canad. Math. Bull. 27 (1984) 48–52, Section 2. [17], ] in the star operation setting. Moreover, we show that D is a Prüfer ⋆-multiplication (respectively, a ⋆-Noetherian; a ⋆-Dedekind) domain if and only if D[X] is a Prüfer [⋆]-multiplication (respectively, a [⋆]-Noetherian; a [⋆]-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel–Popescu localizing systems of finite type on an integral domain D (Problem 45 of [S.T. Chapman, S. Glaz, One hundred problems in commutative ring theory, in: S.T. Chapman, S. Glaz (Eds.), Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, pp. 459–476. [4]]), in terms of multiplicatively closed sets of the polynomial ring D[X].