I have a confusion about one of my question I once asked on this site: In this question I wrote this followings:
Any element, say, $~f(x)+ \langle x^2+x+1, 2\rangle~\in F$ can be written in the form $(ax+b)+\langle x^2+x+1, 2\rangle$, where $a,b \in \mathbb{Z}$ (using Division Algorithm). Now reducing $a,b$ by $\mod 2$ we see that there are $4$ choices of $a,b$.
But is this idea really true? I mean, $\mathbb{Z}[x]$ is not a Euclidean Domain. So I think the Division Algorithm is not applicable there! Am I right? Please mention if there is some silly mistake. Thank you.
Let $R$ be any ring, $g\in R[x]$ monic of degree $m$.
Suppose $f$ has degree $n\ge m$ and its leading coefficient is $a_n$. Then $f_1:=f(x) -a_ng(x)x^{n-m}$ is in the same $g$-coset as $f$ and has strictly smaller degree. Iterate it until the degree becomes smaller than $m$.