On the first page of this preprint, 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?
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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.
answered Aug 12, 2016 at 0:58
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1$\begingroup$ Thanks! He did indeed publish it; see tandfonline.com/doi/abs/10.1080/00927872.2012.753604 $\endgroup$ Commented Aug 12, 2016 at 16:33
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$\begingroup$ I thought so but didn't have access. $\endgroup$ Commented Aug 12, 2016 at 16:44
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$\begingroup$ This is very handy in semigroup theory. It leads to the quickest proof a finite semigroup has a faithful upper triangular representation iff its algebra does. I reproduced this period for my book on semigroup representations. $\endgroup$ Commented Aug 12, 2016 at 16:51