Mathematics – Commutative Algebra
Scientific paper
2007-06-26
Mathematics
Commutative Algebra
Scientific paper
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 \cite[Section 2]{hmm} 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 $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.
Chang Gyu Whan
Fontana Marco
No associations
LandOfFree
Uppers to zero and semistar operations in polynomial rings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Uppers to zero and semistar operations in polynomial rings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Uppers to zero and semistar operations in polynomial rings will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-505618