When do modules mimic arbitrary sets?

ER N. F.

Communications in Algebra, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1080/00927872.2024.2354916
  • Journal Name: Communications in Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: Artinian ring, cogenerating, quasi-Frobenius ring, right chain ring, serial
  • Dokuz Eylül University Affiliated: Yes


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.