Comparison of some purities, flatnesses and injectivities

Abstract

In this paper, we compare $(n,m)$-purities for different pairs of positive integers $(n,m)$. When $R$ is a commutative ring, these purities are not equivalent if $R$ doesn't satisfy the following property: there exists a positive integer $p$ such that, for each maximal ideal $P$, every finitely generated ideal of $R_P$ is $p$-generated. When this property holds, then the $(n,m)$-purity and the $(n,m^{\prime })$-purity are equivalent if $m$ and $m^{\prime }$ are integers $\ge np$. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when $R$ is a semiperfect strongly $\pi$-regular ring. We also compare $(n,m)$-flatnesses and $(n,m)$-injectivities for different pairs of positive integers $(n,m)$. In particular, if $R$ is right perfect and right self $({\aleph }_0,1)$-injective, then each $(1,1)$-flat right $R$-module is projective. In several cases, for each positive integer $p$, all $(n,p)$-flatnesses are equivalent. But there are some examples where the $(1,p)$-flatness is not equivalent to the $(1,p+1)$-flatness.