On minimal extensions of rings

TL;DR

This paper investigates minimal ring extensions, especially those with nonzero ideals intersecting the base ring trivially, and classifies minimal extensions of prime rings, extending previous results in the field.

Contribution

It generalizes the classification of minimal ring extensions to prime rings and explores extensions with specific ideal intersection properties.

Findings

Classified minimal extensions of prime rings.

Identified conditions for extensions with trivial ideal intersection.

Extended previous results on commutative minimal extensions.

Abstract

Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero ideals that intersect $R$ trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions.