Is an irng singly generated as an ideal?

TL;DR

This paper investigates whether finitely generated idempotent rngs (irngs) are singly generated as ideals, extending known results for commutative cases and exploring connections to the Wiegold problem in group theory.

Contribution

It proves that certain classes of irngs, including free and finite irngs, are singly generated as ideals, expanding understanding beyond the commutative case.

Findings

Finitely generated commutative irngs are singly generated.

Free irngs on finitely many idempotents are singly generated.

Finite irngs are also singly generated.

Abstract

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.