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Let $R$ be a commutative Noetherian local ring. We study tensor productsinvolving a finitely generated $R$-module $M$ through the natural action of itsendomorphism ring. In particular, we study torsion properties of self tensorproducts in the case where $\operatorname{End}_R(M)$ has an $R^*$-algebrastructure, and prove that if $M$ is indecomposable, then $M\otimes_{\operatorname{End}_R(M)} M$ must always have torsion in this caseunder mild hypotheses.

Endomorphism algebras over commutative rings and torsion in self tensor products