A Characterization of Rings in which every Semi-Hopfian Module is Noetherian

Abstract

Let  be a ring and  an -module. It is well known that Hopfian modules are semi-Hopfian, but the converse is not always true. For instance, if  is a prime and  is any direct sum of copies of , then  is semi-Hopfian but not Noetherian. In this paper, we introduce and study a new class of rings, called SHN-rings, defined as rings for which every semi-Hopfian module is Noetherian. We investigate the structural properties of SHN-rings and explore their relationships with other well-known classes of rings and modules. Over commutative rings, we establish that SHN-rings coincide with Köthe rings, -rings, pure-semisimple rings, and Artinian principal ideal rings. Furthermore, we show that over commutative SHN-rings, the notions of multiplication modules, cyclic modules, finitely generated modules, and Noetherian modules are equivalent. Additionally, we prove that in the setting of regular rings, semiprime Artinian rings, -rings, and SHN-rings are equivalent.