Some remarks on Prüfer $\star $--multiplication domains and class groups
| dc.creator | Anderson, David F. | |
| dc.creator | Fontana, Marco | |
| dc.creator | Zafrullah, Muhammad | |
| dc.date | 2007-10-26 | |
| dc.date.accessioned | 2026-07-07T08:38:48Z | |
| dc.date.available | 2026-07-07T08:38:48Z | |
| dc.description | Let $D$ be an integral domain with quotient field $K$ and let $X$ be an indeterminate over $D$. Also, let $\boldsymbol{\mathcal{T}}:=\{T_λ\mid λ \in Λ\}$ be a defining family of quotient rings of $D$ and suppose that $\ast $ is a finite type star operation on $D$ induced by $\boldsymbol{\mathcal{T}}$. We show that $D$ is a P$ \ast $MD (resp., P$v$MD) if and only if $(\co_D(fg))^{\ast}=(\co_D(f)\co_D(g))^{\ast}$ (resp., $(\co_D(fg))^{w}=(\co_D(f)\co_D(g))^{w}$) for all $0 \ne f,g \in K[X]$. A more general version of this result is given in the semistar operation setting. We give a method for recognizing P$v$MD's which are not P$\ast $MD's for a certain finite type star operation $\ast $. We study domains $D$ for which the $\ast $--class group $\Cl^{\ast}(D)$ equals the $t$--class group $\Cl^{t}(D)$ for any finite type star operation $\ast $, and we indicate examples of P$v$MD's $D$ such that $\Cl^{\ast}(D)\subsetneq \Cl^{t}(D)$. We also compute $\Cl^v(D)$ for certain valuation domains $D$. | |
| dc.identifier | https://arxiv.org/abs/0710.5018 | |
| dc.identifier | http://arxiv.org/abs/0710.5018 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140862 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13A15; 13C20; 13F05; 13G05 | |
| dc.title | Some remarks on Prüfer $\star $--multiplication domains and class groups | |
| dc.type | text |