Endomorphism algebras over commutative rings and torsion in self tensor products

Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where ${\mathrm{End}}_R(M)$ has an $R^{\ast }$-algebra structure, and prove that if $M$ is indecomposable, then $M{\otimes }_{{\mathrm{End}}_R(M)}M$ must always have torsion in this case under mild hypotheses.