δ-M-Small and δ-Harada Modules

• Published in: Communications in algebra Vol. 36; no. 2; pp. 423 - 433
• Main Authors: Kosan, M Tamer, Ozcan, A Cigdem
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.02.2008
• Subjects: Harada module and ring; Injective module; Lifting module; Noetherian QF-module; Small module
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927870701715779

Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ M -lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained.