We extend some of the results of Agler, Knese, and McCarthy [1] to n-tuples of commuting isometries for n > 2. Let V = (V1, . . . , Vn) be an ntuple of a commuting isometries on a Hilbert space and let Ann(V) denote the set of all n-variable polynomials p such that p(V) = 0. When Ann(V) defines an affine algebraic variety of dimension 1 and V is completely nonunitary, we show that V decomposes as a direct sum of n-tuples W = (W1, . . . , Wn) with the property that, for each i = 1, . . . , n, Wi is either a shift or a scalar multiple of the identity. If V is a cyclic n-tuple of commuting shifts, then we show that V is determined by Ann(V) up to near unitary equivalence, as defined in [1].Theorem 3.4. Suppose V is completely non-unitary, and let I ⊆ Ann(V) be an ideal such that dim Z(I) = 1.(1) I = Ann(V) if and only if I is radical with prime factors I 1 , . . . , I m such that i =j I j Ann(V) for j = 1, . . . , m.(2) If I = Ann(V), then there exist non-zero mutually orthogonal V-invariant subspaces H 1 , . . . , H m of H such that I j = Ann(V|H j ) for each j ∈ {1, . . . , m}.An algebraic variety W in C k is said to be a distinguished variety ifwhere D is the open unit disc in C, T is the unit circle, E is the exterior of the closed unit disc, and exponents indicate Cartesian powers. In the special case that Ann(V) is a prime ideal, the n-tuple V has a particularly simple structure.Theorem 4.4. Suppose V is completely non-unitary. If Ann(V) is a prime ideal and Z(Ann(V)) has dimension 1, then, after a permutation of coordinates, there exists an s ∈ {1, . . . , n} such that (1) V = (V 1 , . . . , V s , λ s+1 I, . . . , λ n I) where V 1 , . . . , V s are shifts and λ s+1 , . . . , λ n are scalars of absolute value 1; and(2) Z(Ann(V)) = W × {(λ s+1 , . . . , λ n )} for some 1-dimensional distinguished variety W ⊆ C s .When Ann(V) is not prime, we have the following result.
