If $g=(f\cdot a+g\cdot b) G-J\cdot g\mod m^{i+1}[x]$ is $g$ a multiple of $G$?

Question

Let $R$ be a local ring with maximal ideal $m$. If we have polynomials

• $f,g\in m^i[x]$
• $a,b, G\in R[x]$
• $J\in m[x]$

such that

$$g=(f\cdot a+g\cdot b) G-J\cdot g\bmod m^{i+1}[x].$$

Then is it correct that $g$ is a multiple of $G$ in $m^{i+1}[x]$? I know we can write

$$(1+J)g=(f\cdot a+g\cdot b)G\bmod m^{i+1}[x]$$

but why does this necessarily mean that $g$ is a multiple of $G$ and not the other way around?

Answer 1

Note that $J \cdot g \in m^{i+1}[x]$, so $g = (f \cdot a + g \cdot b)G \bmod m^{i+1}[x]$.