Communications in Algebra, 2024 (SCI-Expanded)
We study rings whose modules and module homomorphisms display behavior similar to that of sets and their maps. For example, whenever there is an epimorphism (Formula presented.), there is a monomorphism (Formula presented.) (Artinian principal ideal rings (PIR) satisfy this property and its dual for all modules). As a byproduct of this framework, we prove that a ring every factor ring of which cogenerates its cyclic right modules (one-sided version of Kaplansky’s dual rings) is right Artinian and right serial. Consequently, R is an Artinian PIR if and only if every factor ring of R cogenerates its finitely generated right modules. These results can be viewed as partial answers to the CF problem, the FGF problem due to Faith and a question of Faith and Menal on strongly Johns rings. Some known results and the above one yield the following: A ring R is a direct sum of right Artinian right chain rings and Artinian PIR’s if and only if every factor ring of R cogenerates its (uniform) cyclic right modules (with nonzero socle); so, such rings coincide with the right CES-rings of Jain and Lopez-Pérmouth, rings whose factors are right CF and rings that satisfy the above mentioned property for their cyclic right modules A and B. Finally, a ring is either simple Artinian or a right Artinian right chain ring if and only if one of any two cyclic right modules embeds in the other.