Graded Injective Domains
Mike Hensler, Hannah Klawa
TL;DR
The paper develops graded analogs of injective and mated extensions for integral graded domains and connects them to gr-Prüfer domains. It defines graded versions of $i$-extensions, graded-GD, and graded-mated, and establishes that under graded-GD these notions coincide. For integrally closed graded domains, it proves that being gr-Prüfer is equivalent to being gr-mated, and it provides a structural characterization: a graded domain $R$ is a gr-$i$-domain iff $R \subseteq R'$ is a gr-$i$-extension with $R'$ gr-Prüfer. The results extend classical Pap-topological insights to the graded setting and clarify how injective behavior on homogeneous primes interacts with Prüfer-like properties in graded overrings.
Abstract
An integral domain $R$ is an $i$-domain if for every overring $S$ of $R$, $\text{Spec}(S) \rightarrow \text{Spec}(R)$ is injective and is a mated integral if for every overring $S$ of $R$ and prime ideal $P$ of $R$ such that $PS \neq S$, 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.