Graded Injective Domains

TL;DR

This paper introduces graded versions of $i$-domains and mated domains, exploring their properties and relationships with graded Prüfer domains within the context of integral domain theory.

Contribution

It extends classical domain concepts to graded settings and investigates their connections with graded Prüfer domains, providing new insights into graded integral domain structures.

Findings

Defined graded $i$-domains and mated domains.

Established relationships between graded $i$-domains and gr-Prüfer domains.

Explored properties and characterizations of graded mated domains.

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-Pr\"{u}fer domains.