Uppers to Zero and Semistar Operations in Polynomial Rings
نویسنده
چکیده
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 [18, Section 2] 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 [4]), in terms of multiplicatively closed sets of the polynomial ring D[X].
منابع مشابه
Semistar dimension of polynomial rings and Prufer-like domains
Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequal...
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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...
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