FTF rings and Frobenius extensions
type: Item
instance of: scholarly article
title: FTF rings and Frobenius extensions
zbMATH Open document ID: 1005.16003
author: José Gómez-Torrecillas
author: Blas Torrecillas
published in: Journal of Algebra
publication date: 2002-07-04T00:00:00Z
review text: Let \(R\) be a ring with identity element. Then \(R\) is called a left FTF ring if there exists a hereditary torsion theory \(\gamma\) on \(R\)-mod such that a left \(R\)-module \(M\) is \(\gamma\)-torsionfree if and only if \(M\) embeds in a flat left \(R\)-module. Let \(\lambda\) denote the Lambek torsion theory. If \(R\) is left FTF, then \(\gamma\leq\lambda\). If \(P/\lambda(P)\) is torsionless for every finitely presented \(P\in R\)-mod, then \(R\) is called almost absolutely pure. If \(R\) is \(\lambda\)-coherent, then \(R\) is called left almost coherent. The following conditions are equivalent: (1) \(R\) is FTF; (2) \(R\) is almost absolutely pure and almost coherent; (3) direct products of \(E(_RR)\) [and \(E(R_R)\)] are flat left [right] \(R\)-modules; (4) \(R\) is almost coherent and \(E(_RR)\) is flat. If \(S\) is a ring extension of \(R\) such that \(_RS\) and \(S_R\) are flat and if \(S\) is a left FTF ring, then \(R\) is also a left FTF ring. If \(R\) is a left FTF ring and if \(R\to S\) is a ring homomorphism such that \(_RS\) and \(\Hom_R({_RS},R)\) are flat left \(R\)-modules, then \(S\) is a left FTF ring. A ring extension \(R\subseteq S\) is said to be left quasi-Frobenius if \(_RS\) is finitely generated and projective and \(_SS_R\) is isomorphic to a direct summand of a direct sum of finitely many copies of \(_S\Hom_R({_RS},R)\). If \(R\subseteq S\) is a quasi-Frobenius extension, then \(S\) is almost coherent with left (or right) flat-dominant dimension \(\geq 1\) if and only if \(R\) is almost coherent with left (or right) flat-dominant dimension \(\geq 1\). If \(R\subseteq S\) is a quasi-Frobenius extension, then \(S\) has a semi-primary two-sided maximal quotient ring if and only if \(R\) has a semi-primary two-sided maximal quotient ring. If \(R\subseteq S\) is a quasi-Frobenius extension and \(R\) has a QF two-sided maximal quotient ring, so does \(S\).
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 1763563
zbMATH Keywords: left FTF rings
zbMATH Keywords: hereditary torsion theories
zbMATH Keywords: Lambek torsion theory
zbMATH Keywords: almost absolutely pure rings
zbMATH Keywords: left almost coherent rings
zbMATH Keywords: direct products
zbMATH Keywords: flat left modules
zbMATH Keywords: direct sums
zbMATH Keywords: quasi-Frobenius extensions
zbMATH Keywords: maximal quotient rings
reviewed by: Mark L. Teply
MaRDI profile type: Publication
OpenAlex ID: W2140884871
cites work: When is the category of flat modules abelian?
cites work: Rings and modules of quotients
cites work: Torsionfree covers and covers by submodules of flat modules1
cites work: Q3988598
cites work: Strongly graded left FTF rings
cites work: Flat torsionfree modules and QF-\(3\) rings
cites work: On dominant dimension of Noetherian rings
cites work: On Lambek torsion theories
cites work: On Lambek torsion theories. III
cites work: On Lambek torsion theories. II
cites work: Coherence relative to an hereditary torsion theory
cites work: Q4256273
cites work: Q3289058
cites work: Frobenius Extensions of QF-3 Rings
cites work: Autour de la platitude
cites work: Torsion theories and ring extensions
cites work: Semi-primary qf-3 quotient rings
cites work: The descending chain condition relative to a torsion theory
cites work: The endomorphism ring theorem for Frobenius extensions
cites work: Quasi-Frobenius-Erweiterungen
cites work: Q4049223
cites work: On left \(QF-3\) rings
DOI: 10.1006/JABR.2001.9027
description: scientific article; zbMATH DE number 1763563