Characterizations of Graded Prüfer $\star$-Multiplication Domains, II

Abstract

Let $R=\bigoplus _{\alpha \in \Gamma }R_{\alpha }$ 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 $f=f_0+f_1X+\cdots +f_nX^n\in R[X]$, let $\mathcal {A}_f:=\sum _{i=0}^nC(f_i)$. Let $N(\star ):=\{f\in R[X]\mid f\ne 0\text { and }\mathcal {A}_f^{\star }=R^{\star }\}$. In this paper, we study relationships between ideal theoretic properties of ${\text {NA}}(R,\star ):=R[X]_{N(\star )}$ 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 ${\text {NA}}(D,\star )$ is a Prüfer domain and if and only if ${\text {NA}}(R,\star )$ is a Bézout domain. We also determine when ${\text {NA}}(R,v)$ is a PID.