Characterizations of graded Prüfer $\star$-multiplication domains, II

Abstract

Let $R={⨁}_{\alpha \in \Gamma }{R}_{\alpha }$ be a graded integral domain and $\star$ be a semistar operation on $R$. For $a\in R$, denote by $C(a)$ the ideal of $R$ generated by homogeneous components of $a$ and for$f=f_0+f_{1}X+\cdots +f_n{X}^n\in R[X]$, let $\A_f:=\sum_{i=0}^nC(f_i)$. Let $N(\star):=\{f\in R[X]\mid f\neq0\text{and}\A_f^{\star}=R^{\star}\}$. In this paper we study relationships between ideal theoretic properties of $\NA(R,\star):=R[X]_{N(\star)}$ and the homogeneous ideal theoretic properties of $R$. For example we show that $R$ is a graded Prüfer-$\star$-multiplication domain if and only if $\NA(D,\star)$ is a Prüfer domain if and only if $\NA(R,\star)$ is a Bézout domain. We also determine when $\NA(R,v)$ is a PID.

arXiv commentBull. Iranian Math. Soc. to appear