= Monic polynomial division over a ring
{title2=$g=qf+r,\quad\deg r<\deg f$}
If $f(T)$ is monic of degree $k$ over a commutative <ring>, every polynomial has a unique expression $g=qf+r$ with $\deg r<k$. Cancel the highest remaining term successively; the leading coefficient one needs no inversion. Uniqueness follows because multiplication by a monic polynomial raises the degree of every nonzero polynomial by $k$, even when the ring has <zero divisors>. The same argument applies to an even central variable and central coefficients in a <graded commutative algebra>.
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