  Guo and the second author have shown that the closure $[I]$ in the
Drury-Arveson space of a homogeneous principal ideal $I$ in
$\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors
extend this result to the closure of any principal polynomial ideal in the
Bergman space. In particular, the commutators and cross-commutators of the
restrictions of the multiplication operators are shown to be in the Schatten $p
$-class for $p>n$. The same is true for modules generated by polynomials with
vector-valued coefficients. Further, the maximal ideal space $X_I$ of the
resulting $C^\ast$-algebra for the quotient module is shown to be contained in
$Z(I)\cap \partial\mathbb{B}_n$, where $Z(I)$ is the zero variety for $I$, and
to contain all points in $\partial\mathbb{B}_n$ that are limit points of
$Z(I)\cap \mathbb{B}_n$. Finally, the techniques introduced enable one to study
a certain class of weight Bergman spaces on the ball.
