Intersection distributes over equality

Question

Let $N$ be a submodule of an $R$-module $M$ and $I$ an ideal of $R$. Is the equality $$\frac{\bigcap_{n\ge 1}(I^nM+N)}{N}=\bigcap_{n\ge 1}(\frac{I^nM+N}{N})$$ true?

Answer 1

Let $K_n=I^nM+N$. Then you ask if the following holds: $$(\cap_{n\ge 1} K_n)/N=\cap_{n\ge 1}(K_n/N).$$ The inclusion "$\subseteq$" is obvious. Let's show that $(\cap_{n\ge 1} K_n)/N\supseteq \cap_{n\ge 1}(K_n/N)$. Pick an element $\bar x\in \cap_{n\ge 1}(K_n/N)$. Then for every $n\ge 1$ there is $x_n\in K_n$ such that $x-x_n\in N$. Since $N\subseteq K_n$ we get $x\in K_n$ for all $n\ge1$ and we are done.