Uppers to zero and semistar operations in polynomial rings
| dc.creator | Chang, Gyu Whan | |
| dc.creator | Fontana, Marco | |
| dc.date | 2007-06-26 | |
| dc.date.accessioned | 2026-07-07T08:12:25Z | |
| dc.date.available | 2026-07-07T08:12:25Z | |
| dc.description | 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-Prüfer 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 \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Prüfer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Prüfer $[\star]$-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 $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$. | |
| dc.identifier | https://arxiv.org/abs/0706.3761 | |
| dc.identifier | http://arxiv.org/abs/0706.3761 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132447 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13F05; 13A15; 13G05; 13B25 | |
| dc.title | Uppers to zero and semistar operations in polynomial rings | |
| dc.type | text |