Tensor products of faithful modules

Abstract

If $k$ is a field, $A$ and $B$ $k$-algebras, $M$ a faithful left $A$-module, and $N$ a faithful left $B$-module, we recall the proof that the left $A{\otimes }_{k}B$-module $M{\otimes }_{k}N$ is again faithful. If $k$ is a general commutative ring, we note some conditions on $A,$ $B,$ $M$ and $N$ that do, and others that do not, imply the same conclusion. Finally, we note a version of the main result that does not involve any algebra structures on $A$ and $B.$

4 pages. Copy at http://math.berkeley.edu/~gbergman/papers/unpub may be updated more frequently than arXiv copy