Finiteness conditions of $S$-Cohn-Jordan Extensions

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.