On injective modules and annihilators
type: Item
instance of: scholarly article
title: On injective modules and annihilators
zbMATH Open document ID: 0577.16011
author: Roger Yue Chi Ming
published in: Ricerche di Matematica
publication date: 1984-01-01T00:00:00Z
review text: Let A be an associative ring with identity. The ring A is called V-ring iff every simple left A-module is injective. A is called a left PA-ring if for any principal left ideal P of A, there exists a positive integer n such that \(P^ n\) is a left annihilator ideal. A left A-module M is called CE-injective if for any left submodule N, containing a non-zero complement left submodule of M, every left A- homomorphism of N into M extends to an endomorphism of \({}_ AM\). A is called left CE-injective if \({}_ AA\) is CE-injective. A is called ELT if every essential left ideal is twosided. The author considers some generalizations of injectivity. He proves for an ELT ring A that the following conditions are equivalent: (I) A is a left and right selfinjective regular, left and right V-ring of bounded index; (2) A is a semisimple left or right PA, left CE-injective ring; (3) A is a fully left idempotent left CE-injective ring. The author investigates new additional characteristic properties of semisimple Artinian rings [cf. the author, Math. Jap. 19, 173-176 (1974; Zbl 0263.16019)]. Among them the following conditions are equivalent: (I) A is semisimple Artinian; (2) A is ELT such that for any non-zero left ideal I, there exists a positive integer n such that \(I^ n\) is a non- zero projective left annihilator; (3) For any non-zero left ideal I of A, there exists a positive integer n such that \(I^ n\) is a non-zero direct summand of \({}_ AA\); (4) A is either left or right p-injective such that the direct sum of any two CE-injective left A-modules is CE- injective. Considering a ring A with linearly ordered left ideals (LO-ring), the author proves that A is a local left principal ideal quasi-Frobenius ring iff A is a PA, LO-ring with maximum condition on annihilators.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16L60
zbMATH DE Number: 3922896
zbMATH Keywords: simple left A-module
zbMATH Keywords: left PA-ring
zbMATH Keywords: principal left ideal
zbMATH Keywords: complement left submodule
zbMATH Keywords: left CE-injective
zbMATH Keywords: essential left ideal
zbMATH Keywords: generalizations of injectivity
zbMATH Keywords: ELT ring
zbMATH Keywords: V-ring of bounded index
zbMATH Keywords: semisimple Artinian rings
zbMATH Keywords: direct sum
zbMATH Keywords: quasi-Frobenius ring
zbMATH Keywords: maximum condition on annihilators
reviewed by: L.Sh.Ioffe
MaRDI profile type: Publication
Recommended article: Q3682623
Recommended article: Q4340389
Recommended article: Q4862841
Recommended article: Q5289632
Recommended article: Q3024361
Recommended article: Q4017914
Recommended article: Q3774027
Recommended article: Strongly injective modules
Recommended article: Q3758077
Recommended article: Q3977701
description: scientific article; zbMATH DE number 3922896