On the first page of [this preprint](https://math.berkeley.edu/~gbergman/papers/faithful_OX.pdf), I give a proof (a simplification
of an argument suggested by Ken Goodearl) that if *k* is a field,
*A* and *B* are *k*-algebras (not necessarily commutative), *M* is a faithful
left *A*-module, and *N* is a faithful left *B*-module, then the left
*A* ⊗*k B*-module *M* ⊗*k* N is again faithful. It is hard to believe 
this is not classical. Does anyone know a reference?



Is this result on tensor products of faithful modules known?
See Lemma 1.1 of <http://www.math.wisc.edu/~passman/balgebra.pdf> by Passman. I think he published this years later in Communications in Algebra. 


