Let Λ be a left and right noetherian ring. First, for m,n ∈ N ∪ {∞}, we give equivalent conditions for a given Λ-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 Λ-modules which we call upper reducing projective and Gorenstein dimensions. We provide an inequality of upper reducing projective dimension and complexity when Λ 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.
