ON n-SEMIHEREDITARY AND n-COHERENT RINGS

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