Ascending chain conditions on right ideals of semigroups

TL;DR

This paper studies ascending chain conditions on right ideals in semigroups, exploring how these properties are inherited by ideals and ideal extensions, with specific focus on minimal and 0-minimal ideals.

Contribution

It demonstrates that ACCPR is inherited by right and left ideals and provides an example of a right noetherian semigroup with a non-noetherian minimal ideal.

Findings

ACCPR is inherited by right and left ideals

Example of a right noetherian semigroup with a non-noetherian minimal ideal

Analysis of ideal extension behavior

Abstract

We call a semigroup $S$ right noetherian if it satisfies the ascending chain condition on right ideals, and we say that $S$ satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.