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Let $\mathbb{V}$ denote an $n$-tuple of shifts of finite multiplicity, anddenote 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 thereis 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 theequivalence 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 ofpositive integers, where $m$ is the number of irreducible components of thezero set of $\mathrm{Ann}(\mathbb{V})$.

A classification of $n$-tuples of commuting shifts of finite multiplicity