ac.commutative algebra - Generators of ideals in polynomial rings over commutative rings.

This is my first question; I hope it worthy of this awesome forum and its members.

Let $R$ be a commutative ring, perhaps with unit, perhaps not. As usual let $R[x]$
denote the ring of polynomials over $R$, and let $I$ be an ideal in $R[x]$. Let
$D(I)$ be the subset of $I$ consisting of all $f(x) in I$ of minimal degree. It is
well known from elementary algebra that in the event that $R$ is a field, $D(I)$
consists of essentially one element $d(x)$ (up to multiplication by a unit of $R$),
and that $I$ is the principal ideal generated by $d(x)$: $I = (d(x)) = Rd(x)$.
In the case of general commutative $R$, does $D(I)$ generate $I$ in the sense that
any $f(x) in I$ may be written $f(x) = sum_{i = 1}^{m} f_{i}(x)d_i(x)$ with $d_{i}(x)in D(I)$ and $f_{i}(x) in R[x]$? Does the existence of $1 in R$ make any difference?