How to correctly say the following:
Circulant matrix of size $n \times n$ is isomorphic to a ring $\mathbb{Z}[x] / (x^n-1)$
Isomorphic is a strong relationship and may not be suitable here. What I am trying to imply in above sentence is that we can represent a polynomial with integer coefficients as a list of it's coefficients. Then multiply that polynomial by $x^i$ for $0\le i < n$, it would be equal to the rows of cyclic matrix.
You can certainly say that the circulant matrices with integer coefficients form a ring which is isomorphic to the ring $\mathbb Z[x]/(x^n-1)$.
In fact, we can even give an isomorphism: if $X$ is the first orthogonal circulant matrix $$\begin{bmatrix} 0&0&\ldots&0&1\\ 1&0&\ldots&0&0\\ 0&\ddots&\ddots&\vdots&\vdots\\ \vdots&\ddots&\ddots&0&0\\ 0&\ldots&0&1&0 \end{bmatrix}$$ then we can write uniquely any integer circulant matrix as $\sum_{m=0}^{n-1} a_m X^m$ with $a_m \in \mathbb{Z}$.
Using that $X^{m+n} = X^m$ we get that the set of integer circulant matrix is naturally a ring isomorphic to $\mathbb Z[x]/(x^n-1)$.