Exercise 2, § 1

Let f be a polynomial $\neq 0$ in $A[X]$ of degree $n$ and leading coefficient $a_r$. Let M be the submodule of $A[X]$ (qua A-module) consisting of the polynomials in which the coefficient of the term of degree m (for arbitrary $m$ in $\mathbf{N}$) is divisible by $a_0^\mu$, where $\mu = \max(m - n + 1, 0)$; show that for any polynomial g in M there exist two polynomials u, v in $A[X]$ such that $\deg v < n$ and $g = uf + v$.