Let $M$ be an $R$-module and let $x, y \in R$ such that $y$ is not a zero divisor of $M$ and $M / xyM$ has finite length. Show that $l(M/xyM)=l(M/xM)+l(M/yM)$.
In the above, $l$ denotes the lenght of a module. I think I need to find an exact sequence sequence and use the addition theorem for the length of modules, but I am not sure to what sequence I should apply it to. Thx in advance.
I would consider that $xyM$ is a submodule of $yM$, so there is an exact sequence $$ 0\to yM/xyM\to M/xyM\to M/yM\to 0 $$ Now the task becomes to prove that the length of $yM/xyM$ is the same as the length of $M/xM$. Consider the homomorphism $$ f\colon M\to yM/xyM,\qquad f(m)=ym+xyM $$ Compute the kernel of $f$.