Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-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 $D$, in terms of multiplicatively closed sets of the polynomial ring $D[X]$.

CHANG GYU, W., Fontana, M. (2007). Uppers to zero and semistar operations in polynomial rings. JOURNAL OF ALGEBRA, 318, 484-493 [10.1016/j.jalgebra.2007.06.010].

Uppers to zero and semistar operations in polynomial rings

FONTANA, Marco
2007-01-01

Abstract

Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-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 $D$, in terms of multiplicatively closed sets of the polynomial ring $D[X]$.
CHANG GYU, W., Fontana, M. (2007). Uppers to zero and semistar operations in polynomial rings. JOURNAL OF ALGEBRA, 318, 484-493 [10.1016/j.jalgebra.2007.06.010].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/147676
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