Let $��$ be a left and right noetherian ring. First, for $m,n\in\mathbb{N}\cup\{\infty\}$, we give equivalent conditions for a given $��$-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 $��$-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when $��$ 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.

8 pages