Monic polynomial

A monic polynomial has leading coefficient one.

If $f(T)$ is monic of degree $k$ over a commutative ring, every polynomial has a unique expression $g=qf+r$ with $\mathrm{deg}r<k$. 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.

For any commutative ring $A$ and a monic polynomial $F$ of positive degree $D$, monic polynomial division over a ring gives unique representatives of degree less than $D$. Thus the classes of $1,T,\dots ,T^{D-1}$ form a basis of a module over $A$. This is a finite free algebra, hence flat, and the map $A\to A[T]/(F)$ is injective. The monic hypothesis makes leading-degree arguments valid even when $A$ has zero divisors.