Tensor products, projectivity, and subrings

Question

Let $R$ be a unital subring of a unital ring $S$. Let $M$ be an $S$-bimodule that is projective as a right and left module, and let $N$ be a $R$-submodule of $M$, that is again projective as a right and left module. Consider the canonial map $$ N \otimes_R N \to M \otimes_S M, ~~~~~~~ n \otimes n' \mapsto n \otimes n'. $$ Will this map be an embedding in general?

Answer 1

No. Consider the case $R = \mathbb{R}$, $S = \mathbb{C}$, $N = M = \mathbb{C}$. Then $N \otimes_R N = \mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{R}^4$ and $M \otimes_S M = \mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} \cong \mathbb{C}$. The map $N \otimes_R N \to M \otimes_S M$ you described is always $R$-linear, and $\dim_{\mathbb{R}} (\mathbb{R}^4) = 4 > 2 = \dim_{\mathbb{R}}(\mathbb{C})$, so the map will not be injective in this case.