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 EndR(M) End R ( M ) has an R^* R ∗ -algebra structure, and prove that if M is indecomposable, then M ⊗ _EndR(M) M M ⊗ End R ( M ) M must always have torsion in this case under mild hypotheses.