Exercise 111.2.7. Let $(I, \geq )$ be a directed set. Let $A$ be a ring and let $(N_ i, \varphi _{i, i'})$ be a directed system of $A$-modules indexed by $I$. Suppose that $M$ is another $A$-module. Prove that

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} M \otimes _ A N_ i\cong M \otimes _ A \Big( \mathop{\mathrm{colim}}\nolimits _{i\in I} N_ i\Big). \]

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