Let $R$ be a ring and $M$ an $R$-module. Suppose that $J$ : $M=M_0 \supset M_1 \supset \ldots$ is a filtration by submodules. Although the map $M \rightarrow \mathrm{gr}_3 M$ sending $f$ to $\mathrm{in}(f)$ is not a homomorphism of abelian groups, show that either $\mathrm{in}(f)+\mathrm{in}(g)=\mathrm{in}(f+g)$ or $\mathrm{in}(f)+\mathrm{in}(g)=0$. Suppose that $M=R$, and that $J$ is a multiplicative filtration, so that $\operatorname{gr}_3 R$ is a ring. Show that either $\operatorname{in}(f) \operatorname{in}(g)=\operatorname{in}(f g)$ or $\operatorname{in}(f) \operatorname{in}(g)=0$.