For any finitely generated module M with non-zero rank over a commutative
one dimensional Noetherian local domain, the numerical invariant h(M) was
introduced and studied in the author's previous work "Partial Trace Ideals and
Berger's Conjecture". We establish a bound on it which helps capture
information about the torsion submodule of M when M has rank one and it
also generalizes the discussion in the mentioned previous article. We further
study bounds and properties of h(M) in the case when M is the canonical
module ωR​. This in turn helps in answering a question of S. Greco and
then provide some classifications. Most of the results in this article are
based on the results presented in the author's doctoral dissertation "Partial
Trace Ideals, The Conductor and Berger's Conjecture".Comment: 14 pages, Comments are welcom