Finiteness conditions of $S$-Cohn-Jordan Extensions

TL;DR

This paper investigates the conditions under which the $S$-Cohn-Jordan extension of a ring inherits finiteness properties like being noetherian, Bézout, or a principal ideal ring, based on the properties of the original ring.

Contribution

It provides necessary and sufficient conditions linking the finiteness properties of a ring and its $S$-Cohn-Jordan extension under monoid actions.

Findings

Characterization of when $A$ is left noetherian

Criteria for $A$ to be left Bézout

Conditions for $A$ to be a left principal ideal ring

Abstract

Let a monoid $S$ act on a ring $R$ by injective endomorphisms and $A=A(R,S)$ denote the $S$-Cohn-Jordan extension of $R$. Some results relating finiteness conditions of $R$ and that of $A$ are presented. In particular necessary and sufficient conditions for $A$ to be left noetherian, to be left B\'ezout and to be left principal ideal ring are presented.