Comparison of some purities, flatnesses and injectivities

TL;DR

This paper investigates the relationships between various $(n,m)$-purities, flatnesses, and injectivities over different rings, revealing conditions under which these notions are equivalent or distinct, and generalizing existing methods.

Contribution

It generalizes Warfield's methods to compare $(n,m)$-purities and flatnesses, providing new equivalence criteria and exploring these concepts over specific classes of rings.

Findings

$(n,m)$-purities are not equivalent without certain ring properties.

Equivalence of $(n,m)$-purities depends on the property of finitely generated ideals.

Certain rings, like semiperfect strongly $ ext{pi}$-regular rings, exhibit unique purity behaviors.

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 $(n,m')$-purity are equivalent if $m$ and $m'$ are integers $\geq np$. 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.