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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 $M$-projective B$-projective module. The tensor product$M \otimes_A M$is of course again a right free$B$-$B$-bimodule, A$-$A$-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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Free Module with a Projective Modules 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 $M$-projective module.

The tensor product $M \otimes_A M$ is of course again a right free $B$-$B$-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?