Narrow Results

Let $R={\oplus }_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ be an integral domain graded by a nontrivial torsionless grading monoid $\mathrm{\Gamma }$. Among other things, we show that if $\mathrm{\Gamma }\cap (-\mathrm{\Gamma })=\{0\}$, then every nonzero homogeneous ideal of $R$ is invertible if and only if $R_0$ is a field and $R\cong R_0[X]$, where $X$ is an indeterminate over $R_0$, if and only if $R$ is a Dedekind domain, if and only if $R$ is a PID.