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

#### 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]\$.
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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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