Multiplicative closure operations on ring extensions

TL;DR

This paper introduces a new class of multiplicative closure operations on ring extensions, generalizing existing concepts like star and semistar operations, and explores their properties and applications in ring theory.

Contribution

It defines and analyzes multiplicative closure operations on ring extensions, extending the framework of star and semistar operations, and applies these to simplify the study of certain domains.

Findings

Characterization of multiplicative operations on ring extensions.

Behavior of these operations under ring homomorphisms.

Reduction of star operation studies to finite Artinian extensions.

Abstract

Let $A\subseteq B$ be a ring extension and $\mathcal{G}$ be a set of $A$-submodules of $B$. We introduce a class of closure operations on $\mathcal{G}$ (which we call \emph{multiplicative operations on $(A,B,\mathcal{G})$}) that generalizes the classes of star, semistar and semiprime operations. We study how the set $\mathrm{Mult}(A,B,\mathcal{G})$ of these closure operations vary when $A$, $B$ or $\mathcal{G}$ vary, and how $\mathrm{Mult}(A,B,\mathcal{G})$ behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.