Huneke Wiegand conjecture of rank one with the change of rings
Let (R,\fkm) be a Cohen Macaulay local ring of dimension one and assume that $R$ possesses a canonical ideal \rmK_R$ Let $I$ be a faithful ideal of $R$ We explore the problem of when $I \otimes_RI^\vee} is torsionfree, where $I^\vee} \Hom_R(I, \rmK_R) If the multiplicity of $R$ with respect to \fkm$ is at most 4, or if \fkm \bar{R} \subseteq R$ where \bar{R} stands for the integral closure of $R$ in its total ring of fractions, then $I \cong R$ or $I\cong \rmK_R$ as an $R$ module, once $I\otimes_RI^\vee} is torsionfree. Applying this result together with other ones to Gorenstein numerical semigroup rings $R k[[t^{a_1} t^{a_2} \ldots, t^{a_{\ell} over a field $k$ (here $t$ denotes an indeterminate and $0 a_1 a_2< \cdots <a_{\ell} are integers with \mathrm{gcd}(a_1, a_2, \ldots, a_{\ell} 1$ we will show in several cases that $I$ is a principal ideal, once $I \otimes_R\Hom_R(I,R) is torsionfree, provided $I$ is generated by monomials in $t$ A higher dimensional assertion is discussed.