I am working through Eisenbud's Commutative Algebra, and in Chapter $5$ he defines the following map. Say we have a filtration of modules ${\cal F}:M=M_0\supset M_1\supset\cdots$. Then for $f\in M$, let $n$ be the greatest natural number so that $f\in M_n$. He defines $\operatorname{in}(f)=f\pmod {M_{n+1}}\in M_n/M_{n+1}$ (and if $f$ is in all $M_i$ then define $\operatorname{in}(f)=0$).
I am trying to prove some properties of the map for my homework, but I am trying to make sense of the expression: $\operatorname{in}(f)+\operatorname{in}(g)$, because these two terms are in different sets, e.g. how can we add $x\in M_1/M_2$ with $y\in M_4/M_5$?