Computing the closure of a support

TL;DR

This paper explicitly characterizes the Zariski closure of the support of an R-module over a commutative ring, explores its algebraic properties, and discusses applications to ring extensions and ideal theory.

Contribution

It provides an explicit description of the radical ideal defining the support closure and analyzes its behavior under algebraic operations, with applications to ring extension ideals.

Findings

Explicit form of the radical ideal for support closure

Behavior of the support closure under algebraic operations

Applications to ring extensions and ideal theory

Abstract

When $E$ is an $R$-module over a commutative unital ring $R$, the Zariski closure of its support is of the form $\mathrm V(\mathcal O(E))$ where $\mathcal O(E)$ is a unique radical ideal. We give an explicit form of $\mathcal O(E)$ and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.