Given an ideal in a ring $I\subset R$, we have the associated graded ring $$\mathrm{gr}_I(R) = \bigoplus_{i=0}^{\infty}I^i/I^{i+1}$$ One can form the initial form map $\iota: R \to \mathrm{gr}_I(R)$, sending $R \ni r \mapsto \bar{r} \in I^i/I^{i+1}$,
where $I^i$ is the smallest power of $I$ containing $r$.
This map is not in general a homomorphism, but we do have that either $\iota(f+g) = \iota(f) + \iota(g)$ or $\iota(f+g) = 0$.
I stated this for rings, but it may be better to state for modules.
For reference you can consult either https://en.wikipedia.org/wiki/Associated_graded_ring or Eisenbud's Commutative Algebra (I think that was where I read about it).