On rings admitting nonzero homomorphisms between non-projective modules


Alagöz Y., Durğun Y., İzci K.

COMMUNICATIONS IN ALGEBRA, vol.51, no.4, pp.1614-1622, 2023 (SCI-Expanded)

  • Publication Type: Article / Article
  • Volume: 51 Issue: 4
  • Publication Date: 2023
  • Doi Number: 10.1080/00927872.2022.2139970
  • 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
  • Page Numbers: pp.1614-1622
  • Çukurova University Affiliated: Yes

Abstract

In a recent paper, Türkoğlu have studied the rings admitting nonzero homomorphisms between any two non-projective modules (property (T)). His work raises the following question: What are the rings with nonzero maps between non-projective right modules and singular simple modules? As a generalization of the property (T), we consider two families of rings: those rings with nonzero maps from any non-projective module to any singular simple module (property (T1)) and those rings with nonzero maps from any singular simple module to any non-projective module (property (T2)). Complete characterizations of both classes of rings are obtained and it is shown, in particular, that the rings satisfying property (T) are precisely those rings satisfying both properties (T1) and (T2). We show that R satisfies (T) if and only if R has a unique singular simple right R-module, and R is either two-sided Artinian hereditary serial or right completely coretractable. Furthermore, we prove that a commutative ring R has the property (T) if and only if there is a ring decomposition 𝑅𝐴×𝐵, where A is semisimple Artinian and B satisfies one of the following conditions: (1) B is a semi-Artinian max ring with unique singular simple right R-module. (2) B is an Artinian hereditary serial ring with unique singular simple right R-module.