Can you instantiate Ring-LWE with coefficients from a prime-power field?

Generally, we instantiate Ring-LWE with the polynomial ring $R = \mathbb{F}_q\ /\ (X^N+1)$ for prime $q$ and some power-of-two $N$.

Can we instead do Ring-LWE over the ring $R = \mathbb{F}_q\ /\ (X^N+1)$, where $q$ can be any prime power? Basically, this would mean the coefficients of ciphertexts are elements of $GF(q)$.

(Also, is this an unusual choice, or covered by existing literature that I'm not aware of?)

Yes, it is possible to instantiate Ring-LWE with the polynomial ring R=F_q/ (X_N+1), where q is a prime power and N is a power-of-two. in this case, Ring-LWE is known as "Finite Field Ring-LWE" or "FF-Ring-LWE".

In FF-Ring-LWE, the coefficients of the ciphertexts are elements of the finite field GF(q), which is the field of order q. The arithmetic operations in GF(q) are similar to those in Zq, but with some differences due to the fact that GF(q) is a field rather than a ring.

However, the choice of q can have an impact on the security of the scheme. In particular, if q is too small, then the scheme may be vulnerable to certain attacks, such as lattice reduction algorithms. On the other hand, if q is too large, then the scheme may be less efficient and require larger key sizes.