  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 $\operatorname{End}_R(M)$ has an $R^*$-algebra
structure, and prove that if $M$ is indecomposable, then $M
\otimes_{\operatorname{End}_R(M)} M$ must always have torsion in this case
under mild hypotheses.

Commutative Algebra;Rings and Algebras