What is the quotient of a quotient module?

Question

Say you have $N=M\big/K$, where $M$ is an $R$-module and $K$ lies inside $M$. When I take $N\big/IN$, with $I$ being an ideal in $R$, is that equal to $$M\big/\big(IM + K)\ ?$$ How do you find that out ?

Answer 1

$N/IN=\frac{M/K}{I(M/K)}=\frac{M/K}{(IM+K)/K}\cong\frac{M}{IM+K}$

The first equality is just unpacking the notation you chose to eliminate $N$.

The second equality computes $I(M/K)$, which by definition must be of the form $A/K$ where $A$ contains $IM$ and $K$, and the submodule that fits the bill then is $IM+K$.

The isomorphism is simply the "third isomorphism theorem."

Perhaps you should not say that it is equal to because as sets they are not equal. But they are the same up to isomorphism of modules, if that is what you want.