Finiteness conditions of \(S\)-Cohn-Jordan extensions. (Q2909793)

This paper is a continuation of the investigations of \textit{D. A. Jordan} [J. Lond. Math. Soc., II. Ser. 25, 435-448 (1982; Zbl 0486.16002)] and the author [Commun. Algebra 35, No. 3, 725-746 (2007; Zbl 1143.16025)] on the relationship between various algebraic properties of \(R\) and that of an \(S\)-Cohn-Jordan extension \(A(R;S)\) of the ring \(R\). Here \(S\) is a multiplicative monoid that acts on a ring \(R\) by injective endomorphisms and \(A(R;S)\) is a minimal over-ring of \(R\) for which the action of \(S\) on \(R\) extends to the action of \(S\) on \(A(R;S)\) by automorphism.NEWLINENEWLINE The objective here is to study finiteness conditions on \(A(R;S)\) in terms of properties of the ring \(R\) and the action of \(S\). In order to do so, the author firstly determines a correspondence between the left ideals of \(A(R;S)\) and certain admissible sets of left ideals in \(R\). Subsequently necessary and sufficient conditions for \(A(R;S)\) to be Noetherian, or a left principal ideal ring or a left Bézout ring, respectively, are given. Examples are provided to show that the relationships between \(R\) and \(A(R;S)\) are not always straightforward. For example, there are principal ideal domains \(R\) for which \(A(R;S)\) is not Noetherian.