Composition series relative to a module
type: Item
instance of: scholarly article
title: Composition series relative to a module
zbMATH Open document ID: 0549.16016
DOI: 10.1016/0022-4049(85)90027-1
author: Tatsuo Izawa
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: The results of \textit{O. Goldman} [J. Algebra 35, 308-341 (1975; Zbl 0313.16002)] for composition series relative to an hereditary torsion theory are generalized to the non-hereditary case. In particular, for a module \(U_ R\), a module \(M_ R\) is U-torsion if \(Hom_ R(M_ R,U_ R)=0\); \(N_ R\) is U-cocritical if \(N_ R\) is U-torsionless and any proper homomorphic image of \(N_ R\) is U-torsion. Then a U-composition series can be defined in the obvious way, and the analogs for the usual existence, refinement, and uniqueness theorems are proved. If U is quasi- injective and \(S=End_ R(U)\), then there is a natural Galois type relationship between a U-composition series for \(M_ R\) and a composition series of \({}_ SHom_ R(M_ R,U_ R)=_ SM^*\). Applications are made to the study of endomorphism rings and the structure of \({}_ SM^*\).
reviewed by: Mark L. Teply
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 3875479
zbMATH Keywords: composition series
zbMATH Keywords: torsion theory
zbMATH Keywords: cocritical
zbMATH Keywords: torsionless
zbMATH Keywords: quasi- injective
zbMATH Keywords: endomorphism rings
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90027-1
OpenAlex ID: W2035177053
cites work: On M-projective and M-injective modules
cites work: Rings satisfying a minimum condition on principal ideals.
cites work: Injective Modules and Injective Quotient Rings
cites work: Q4055809
cites work: Elements of noncommutative arithmetic. I
cites work: On endomorphism rings of Noetherian quasi-injective modules
cites work: The descending chain condition relative to a torsion theory
cites work: Coherent Rings and Fp -Injective Modules
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description: scientific article; zbMATH DE number 3875479