  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.
