<s>[INST] Using ring R=$\mathbb{Z}[X]/(X^n+1)$ Currently studying Ring-LWE,here, and having trouble grasping a concrete definition of what this means, as a ring. My understanding is that everything in this ring is an integer polynomial, and that it must be reduced $X^n+1$, is this correct? Essentially working modulo $X^n+1$.
It then goes on to work in quotient group $R_q=R/qR$. When in this quotient group, would all elements have already been reduced $X^n+1$ and finally then reduced mod $q$?
 [/INST] The polynomials are just multiplied as usual but any term $X^N$ is replaced by $-1$, $X^{n+1} \equiv -X$ etc. , as we consider $X^n+1$ to be equivalent to $0$. 
All coefficients of those polynomials lie in $\mathbb{Z}_q$, so whenever they go above $q$, reduce them. 
So a simple example: $n=3$, $q=3$, then $$(1+X+X^2)(1+2X+2X^2)= 1 + 2X + 2X^2 + X + 2X^2 + 2X^3 + X^2 + 2X^3 + 2X^4=1 + 3X + 5X^2 +4X^3 + 2X^4 \equiv 1+ 2X^2 + 4(-1) + 2(-X) \equiv (1-4) - 2X +2X^2 \equiv X +2X^2$$
for example (modulo calculation errors, working from screen)
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