Uppers to zero and semistar operations in polynomial rings.pdfVIP

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Uppers to zero and semistar operations in polynomial rings.pdf

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Uppers to zero and semistar operations in polynomial rings

a r X i v : 0 7 0 6 .3 7 6 1 v 1 [ m a t h .A C ] 2 6 J u n 2 0 0 7 UPPERS TO ZERO AND SEMISTAR OPERATIONS IN POLYNOMIAL RINGS GYU WHAN CHANG AND MARCO FONTANA Abstract. 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-Pru?fer 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 [18, Section 2] in the star operation setting. More- over, we show that D is a Pru?fer ?-multiplication (resp., a ?-Noetherian; a ?-Dedekind) domain if and only if D[X] is a Pru?fer [?]-multiplication (resp., a [?]-Noetherian; a [?]-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu local- izing systems of finite type on an integral domain D (Problem 45 of [4]), in terms of multiplicatively closed sets of the polynomial ring D[X]. 1. Introduction and background results Let D be an integral domain with quotient field K. Let F (D) denote the set of all nonzero D–submodules of K and let F (D) (resp., f(D)) be the set of all nonzero fractional (resp., finitely generated fractional) ideals of D. Following Okabe-Matsuda [22], a semistar operation on D is a map ? : F (D) → F (D), E 7→ E?, such that, for all x ∈ K, x 6= 0, and for all E,F ∈ F (D), (a) (xE)? = xE?; (b) E ? F implies E? ? F ?; (c) E ? E? and E?? := (E?)? = E?. A (semi)star operation is a semistar operation that, restricted to F (D), is a star operation (in the sense of [14, Section 32]). It is easy to see that a semistar operation ? on D is a (semi)star operation if and only if D? = D. If ? is a semistar operation on D, then we can consider a map ? f : F (D) → F (D) defined by E?f := ? {F ? | F ∈ f(D) and F ? E}, for each E ∈ F (D). It is easy to see that ? f is a semistar ope