Uppers to Zero in Polynomial Rings and Prüfer-like Domains

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e, its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content cD(g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with cD(g) v = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasiPrüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this paper, given a semistar operation ⋆ in the sense of Okabe-Matsuda, we introduce the ⋆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains. introduction and background results Gilmer and Hoffmann characterized Prüfer domains as those integrally closed domains D, such that the extension of D inside its quotient field is a primitive extension [28, Theorem 2]. (Relevant definitions and results are reviewed in the sequel.) Primitive extensions are strictly related with relevant properties of the prime spectrum of the polynomial ring. In particular, from the previous characterization it follows that a Prüfer domain is an integrally closed quasi-Prüfer domain (i.e., an integral domain such that each prime ideal of the polynomial ring contained in an extended prime is extended [5]) [19, Section 6.5]. A quasi-Prüfer domain D can be characterized by the fact that each upper to zero Q in D[X ] contains a polynomial g ∈ D[X ] with content cD(g) = D (Theorem 1.1). On the other hand, a “weaker” version of the last property can be used for characterizing upper to zero that are maximal t-ideals in the polynomial ring. Recall that D is called a UMt-domain (UMt means “upper to zero is a maximal t-ideal”) if every upper to zero in D[X ] is a maximal t-ideal [42, Section 3] and this happens if and only if each upper to zero in D[X ] contains a nonzero polynomial g ∈ D[X ] with cD(g) = D [21, Theorem 1.1]. Using the previous observations, the notions of UMt-domain and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. More precisely, given a semistar operation ⋆ in the sense of Okabe-Matsuda [52], we introduce in a natural way the ⋆-quasi-Prüfer domains