Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M \otimes_R S$ is injective for any right $R$-module $M$, and $N \to S \otimes_R N$ is injective for any left $R$-module $N$.

Is it then true that $M \otimes_R N \to M\otimes_R S \otimes_R N$ is injective for any pair $(M, N)$ of a left and a right $R$-module? This seems way too optimistic, but I can't seem to find a counterexample.

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M \otimes_R S$ is injective for any right $R$-module $M$, and $N \to S \otimes_R N$ is injective for any left $R$-module $N$.

Is it then true that $M \otimes_R N \to M\otimes_R S \otimes_R N$ is injective for any pair $(M, N)$ of a right and a left $R$-module? This seems way too optimistic, but I can't seem to find a counterexample.