Graded Injective Domains

An integral domain $R$ is an $i$-domain if for every overring $S$ of $R$, $\text{Spec}(S)\to \text{Spec}(R)$ is injective and is a mated integral if for every overring $S$ of $R$ and prime ideal $P$ of $R$ such that $PS\mathrm{\ne }S$, there exists exactly one prime ideal $Q$ of $S$ such that $Q\cap R=P$. In this paper, we explore graded notions of $i$-domains and mated domains and their connection with gr-Prüfer domains.