Exercise 111.15.3. Let $R$ be a ring. Let $f = a_ d x^ d + a_{d - 1} x^{d - 1} + \ldots + a_0 \in R[x]$. (As usual this notation means $a_0, \ldots , a_ d \in R$.) Let $g \in R[x]$. Prove that we can find $N \geq 0$ and $r, q \in R[x]$ such that

$a_ d^ N g = q f + r$

with $\deg (r) < d$, i.e., for some $c_ i \in R$ we have $r = c_0 + c_1 x + \ldots + c_{d - 1}x^{d - 1}$.

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