Is the quotient ring $\mathbb{Z}_4[x]/$ a vector space? I know that if p is a prime number, then the quotient ring $$\frac {\mathbb{Z}_p[x]}{ \langle x^n-1 \rangle }$$ is a vector space over $\mathbb Z_p$. What if p is not a prime? Certainly, $\mathbb{Z}_m[x]$ is not a Euclidean domain while m is not prime. Is it necessary to be a Euclidean domain in order to obtain a vector space?

The problem is that you want the vector space axioms to hold. In particular, you want to be able to multiply by scalars from a field. If $p$ is not prime, then it is not obvious which field should act on the ring.
