\"Uber die maximalen Ideale des Quotientenringes $R_{\mathfrak{G}}$

TL;DR

This paper characterizes simple torsion-free and divisible Artinian modules over a commutative Noetherian local ring with respect to a Gabriel topology, linking maximal elements of the spectrum to maximal ideals of a quotient ring.

Contribution

It provides a complete description of simple modules related to a Gabriel topology, connecting module theory with the structure of the quotient ring $R_{rak{G}}$.

Findings

Identification of simple $rak{G}$-torsion free modules

Classification of simple $rak{G}$-divisible Artinian modules

Maximal ideals of $R_{rak{G}}$ correspond to elements of $rak{G}^*$

Abstract

Let $R$ be a commutative Noetherian local ring, $\mathfrak{G}$ a Gabriel topology on $R$, and $\mathfrak{G}^\ast$ the set of all maximal elements of Spec($R)\backslash \mathfrak{G}$. We determine all simple $\mathfrak{G}$-torsion free $R$-modules $M$, as well as all simple $\mathfrak{G}$-divisible Artinian $R$-modules $N$. A central role is played by the set $\mathfrak{G}^{\ast}$: Its elements correspond exactly to the maximal ideals of the quotient ring $R_{\mathfrak{G}}$, if $\mathfrak{G}$ is perfect.