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.