The radical-annihilator monoid of a ring

TL;DR

This paper introduces a monoid generated by radical and annihilator operations on ring ideals, characterizing certain classes of rings and analyzing their algebraic structure.

Contribution

It defines and studies the radical-annihilator monoid in ring theory, providing characterizations for semiprime and dual rings and analyzing specific local rings.

Findings

Semiprime rings characterized by their monoid structure

Commutative dual rings characterized by their monoid

Monoids for local zero-dimensional rings determined

Abstract

Kuratowski's closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by "radical" and "annihilator" maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.