  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 $\omega_R$. 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". 
Partial Trace Ideals, Torsion and Canonical Module