FINITENESS CONDITIONS OF S-COHN-JORDAN EXTENSIONS

Let a monoid S act on a ring R by injective endomorphisms and A(R; S) denote the S-Cohn-Jordan extension of R. Some results relating finiteness conditions of R and that of A(R; S) are presented. In particular necessary and sufficient conditions for A(R; S) to be left noetherian, to be left Bezout and to be left principal ideal ring are presented. This also offers a solution to Problem 10 from [On S-Cohn-Jordan extensions, in Proc. 39th Symp. Ring Theory and Representation Theory, Hiroshima, ed. M. Kutami (Hiroshima Univ., Japan, 2007), pp. 30-35].