  Let $\mathbb{V}$ denote an $n$-tuple of shifts of finite multiplicity, and
denote by $\mathrm{Ann}(\mathbb{V})$ the ideal consisting of polynomials $p$ in
$n$ complex variables such that $p(\mathbb{V})=0$. If $\mathbb{W}$ on
$\mathfrak{K}$ is another $n$-tuple of shifts of finite multiplicity, and there
is a $\mathbb{W}$-invariant subspace $\mathfrak{K}'$ of finite codimension in
$\mathfrak{K}$ so that $\mathbb{W}|\mathfrak{K}'$ is similar to $\mathbb{V}$,
then we write $\mathbb{V}\lesssim \mathbb{W}$. If $\mathbb{W}\lesssim
\mathbb{V}$ as well, then we write $\mathbb{W}\approx \mathbb{V}$.
  In the case that $\mathrm{Ann}(\mathbb{V})$ is a prime ideal we show that the
equivalence class of $\mathbb{V}$ is determined by $\mathrm{Ann}(\mathbb{V})$
and a positive integer $k$. More generally, the equivalence class of
$\mathbb{V}$ is determined by $\mathrm{Ann}(\mathbb{V})$ and an $m$-tuple of
positive integers, where $m$ is the number of irreducible components of the
zero set of $\mathrm{Ann}(\mathbb{V})$.
