2007/06/26 by Gyu Whan Chang, Chang, Gyu Whan, Marco Fontana +1
Mathematics · #13A15 #13B25 #13F05 #13G05 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.0706.3761
openalex publication_date 2007/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \cite[Section 2]hmm in the star operation setting. Moreover, we show that D is a Prüfer ⋆-multiplication (resp., a ⋆-Noetherian; a ⋆-Dedekind) domain if and only if D[X] is a Prüfer [⋆]-multiplication (resp., 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 \citecg), in terms of multiplicatively closed sets of the polynomial ring D[X].