ON n-SEMIHEREDITARY AND n-COHERENT RINGS

Year 2007, Volume: 1 Issue: 1, 1 - 10, 01.06.2007

Xiaoxiang Zhang Jianlong Chen

Abstract

Let R be a ring. For a fixed positive integer n, R is said to
be left n-semihereditary in case every n-generated left ideal is projective. R
is said to be weakly n-semihereditary if each n-generated left (and/or right)
ideal is flat. Some properties of n-semihereditary rings, respectively, weakly
n-semihereditary rings and n-coherent rings are investigated. It is also proved
that R is left n-semihereditary if and only if it is left n-coherent and weakly
n-semihereditary, if and only if the ring of n × n matrices over R is left 1-
semihereditary if and only if the class of all n-flat right R-modules form the
torsion-free class of a torsion theory. Some known results are extended or
obtained as corollaries

Keywords

(weakly) n-semihereditary ring, n-coherent ring

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• Xiaoxiang Zhang and Jianlong Chen
• Department of Mathematics, Southeast University
• Nanjing 210096, P. R. China
• e-mail: [e-mail] (X. Zhang), [e-mail] (J. Chen)