Subquotients of Module

Question

In Rotman's book on homological algebra (page 625), he says that if you have modules $Y\subset X\subset Z$ with $X/Y=Z$ then $Y=0$ and $X=Z$. It's not clear what he means by $X/Y=Z$ in the first place, but I can only assume he means isomorphic, although there then seems to be fairly easy counter-examples: $\mathbb{Z}^{\mathbb{4N}}\subset \mathbb{Z}^{\mathbb{2N}}\subset \mathbb{Z}^{\mathbb{N}}$, where for example $\mathbb{Z}^{\mathbb{2N}}=\{(0,x_{1},0,x_{2},0,..): x_{i}\in \mathbb{Z}\}$, since $\mathbb{Z}^{\mathbb{2N}}/\mathbb{Z}^{\mathbb{4N}}=\mathbb{Z}^{\mathbb{N}}$ via a bijection between $\mathbb{2N}\setminus\mathbb{4N}$ and $\mathbb{N}$. Can anyone correct me, or is this indeed a mistake?

Answer 1

What Rotman wants to prove is that if $E^r= E^{r+1}$ in a spectral sequence, then $Z^{r+1} = Z^r$ and $B^{r+1} = B^r$. Remember that $E^{r+1}$ is computed as the homology of $E^r$ with respect to a differential. What Rotman is saying here is: the differential is zero iff $E^r=E^{r+1}$, iff the cycles (resp. boundaries) at stage $r$ are the same as those at stage $r+1$.

Indeed, note that if $E^r=E^{r+1}$ then there is no differential, so the cycles are equal, and now you want to prove that if $A \subseteq B \subseteq C$ and the natural map $C/B \to C/A$ is an isomorphism, then $A=B$. The converse is obvious.