Rings whose proper cyclic modules are artinian
by
Ahmad Shamsuddin
American University of Beirut

Fuchs in his book Abelian Groups stated that a ring R for which all proper cyclic left R-modules are artinian is left noetherian. Camillo and Krause pointed out a gap in the proof of Fuchs's theorem and turned that theorem into a question. It is not difficult to show that this is equivalent to asking whether rings with left Krull dimension 1 are noetherian. We discuss this problem, mainly in the case when the Jacobson radical of R is non-zero. For example, we show that all epimorphisms of cyclic modules in this case are injective, and in case the ring is local, we show that it is left repetetive.

Date received: October 7, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-02.