We give a positive answer to the Huneke-Wiegand Conjecture for monomial ideals over free numerical semigroup rings, and for two generated monomial ideals over complete intersection numerical semigroup rings.It is often the case that open problems in ring theory remain difficult when specialized to numerical semigroup rings. In such instances it may be beneficial to gain perspective on the problem by trying to tackle its number theoretic analog. Since the integral closure of a numerical semigroup ring is just the polynomial ring in one variable, this perspective seems all the more reasonable for problems where the case for integrally closed rings is much easier to solve.If R is a one-dimensional integrally closed local domain and M is a finitely generated torsion-free R-module, then M is free if and only if M ⊗ R Hom(M, R) is torsion-free. This follows from either [1, 3.3] or from the structure theorem for modules over a principle ideal domains. C. Huneke and R. Wiegand have conjectured that this property holds for all one-dimensional Gorenstein domains.Conjecture 1. [7,[473][474] Let R be a one-dimensional Gorenstein domain and let M be a non-zero finitely generated R-module, which is not projective. Then M ⊗ R Hom R (M, R) has a non-trivial torsion submodule.