  Let $\Lambda$ be a left and right noetherian ring. First, for
$m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given
$\Lambda$-module to be $n$-torsionfree and have $m$-torsionfree transpose.
Using them, we investigate totally reflexive modules and reducing Gorenstein
dimension. Next, we introduce homological invariants for $\Lambda$-modules
which we call upper reducing projective and Gorenstein dimensions. We provide
an inequality of upper reducing projective dimension and complexity when
$\Lambda$ is commutative and local. Using it, we consider how upper reducing
projective dimension relates to reducing projective dimension, and the complete
intersection and AB properties of a commutative noetherian local ring.
