# Free Module with a Projective Sub- Module, and Tensor Products

Let us consider a unital algebra $A$, with a subalgebra $B \subseteq A$, along with an $A$-$A$-bimodule $M$ which is free as a right module, and a subspace $N$ (with respect to the action of the field coming from the unit of $A$) such that $BNB \subseteq N$, and $N$ is a right $B$-projective module.

The tensor product $M \otimes_A M$ is of course again a right free $A$-$A$-bimodule, and the tensor product $N \otimes_B N$ is again projective as a right $B$-module. What I would like to know is whether the canonical insertion of $N \otimes_B N$ into $M \otimes_A M$ is an embedding?

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It seems to me that some variables are meant to be other variables ... –  Martin Brandenburg Jun 20 '12 at 16:59
Yes, sorry about that. It's fixed now. –  Milan Bernolak Jun 20 '12 at 17:05

No. Take $N=M=A$, where $A$ is any non-trivial algebra over a field $k$, and $B=k$.