MMN-5249
Homological characterizations of G-Krull domains and G-Dedekind domains
Dechuan Zhou; Hwankoo Kim; Kui Hu;Abstract
Gorenstein Krull domains (G-Krull domains) are defined as domains $R$ that satisfy the following three conditions: (1) For each prime ideal $\fkp$ of $R$ of height one, $R_\fkp$ is a Gorenstein ring. (2) $R = \bigcap R_\fkp$, where $\fkp$ ranges over all prime ideals of $R$ of height one. (3) Any nonzero element of $R$ lies in only a finite number of prime ideals of height one. In this paper, we aim to characterize G-Krull domains from the perspective of Gorenstein homological algebra, similar to Gorenstein Dedekind domains (G-Dedekind domains). To achieve this objective, we introduce the notion of $w$-locally Gorenstein projective modules (G-projective modules). An $R$-module $M$ is called $w$-locally Gorenstein projective if $M_\mathfrak{m}$ is G-projective for any maximal $w$-ideal $\mathfrak{m}$ of $R$. We show that a domain $R$ is G-Krull if and only if $R$ is a strong Mori domain and every $w$-ideal of $R$ is $w$-locally G-projective. Additionally, we establish that a domain $R$ is G-Dedekind if and only if $R$ is a Noetherian domain and every maximal ideal of $R$ is G-projective.
Vol. 27 (2026), No. 1, pp. 455-464