On Lambek torsion theories
type: Item
instance of: scholarly article
title: On Lambek torsion theories
zbMATH Open document ID: 0787.16026
author: Mitsuo Hoshino
published in: Osaka Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Let \(R\) be a ring with identity, and let \(\varepsilon_ X: X\to X^{**}\) be the canonical evaluation map from \(X\) to its double dual. Recall that the Lambek torsion theory \(\tau\) is defined via \(\text{Hom}_ R(\tau(X),E(R)) = 0\) and \(E(R)\) cogenerates \(X/\tau(X)\). The following statements are equivalent: (1) \(\tau(X) = \text{ker }\varepsilon_ X\) for every finitely presented \(X \in R\)-mod; (2) \(\tau(M) = \text{ker }\varepsilon_ M\) for every finitely presented \(M\in \text{mod-}R\); (3) \(f^{**}\) is monic for each \(f: X\to Y\) with \(_ RX\) finitely generated and \(_ RY\) finitely presented; (4) \(g^{**}\) is monic for each \(g: M\to N\) with \(M_ R\) finitely generated and \(N_ R\) finitely presented. If \(R\) is right coherent, then the following statements are equivalent: (1) \(E(_ RR)\) is flat; (2) there is a faithful, injective, flat left \(R\)-module; (3) \(\tau(X)=\text{ker }\varepsilon_ X\) for every finitely presented \(X\in R\)-mod. If \(R\) is right noetherian, then \(E(_ RR)\) is flat if and only if every finitely generated submodule of \(E(R_ R)\) is torsionless. If \(R\) is left and right coherent, then the following statements are equivalent: (1) weak \(\dim E(_ RR) \leq 1\); (2) \(\varepsilon_ X\) is an essential monomorphism for every torsionless \(X\in R\)-mod; (3) \(\text{Ext}^ 1_ R(-,R)^*\) vanishes on \(R\)-mod. The vanishing of \(\text{Ext}^ 1_ R(- ,R)^*\) on \(R\)-mod is not symmetrical. The author also notes that S. Takashima has been able to weaken the overall hypotheses in the second and third results stated in this review.
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16P70
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 219886
zbMATH Keywords: torsionless modules
zbMATH Keywords: Lambek torsion theory
zbMATH Keywords: finitely generated
zbMATH Keywords: finitely presented
zbMATH Keywords: right coherent
zbMATH Keywords: faithful, injective, flat left \(R\)-module
zbMATH Keywords: essential monomorphism
reviewed by: Mark L. Teply
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 219886