What is $\mathbb{Z_p}$ when $p$ is polynomial? --- Short --
What is the ring  $\frac{\mathbb{Z}_p[x]}{x^n + 1}$ if we allow $p$ to be a polynomial?
-- Contextualized question --
In the paper Efficient Integer Encoding for Homomorphic Encryption via Ring Isomorphisms, a cryptographic scheme that uses a ring $R_p = \frac{\mathbb{Z}_p[x]}{x^n + 1}$, where $p$ is a integer, is modified to create a encoding method...
So, the authors choose $p = x - a$, for some $a \in \mathbb{Z}$ and say that 
they will work with the quotient ring $R_{x -a} = \frac{\mathbb{Z}[x]}{(x^n + 1, x - a)}$
First of all, I am not sure that $R_{x -a} = \frac{\mathbb{Z}[x]}{(x^n + 1, x - a)}$ is the samething of $R_{x -a} = \frac{\mathbb{Z}_{x - a}[x]}{x^n + 1}$, but I think it is the case.
Second, I don't know what the authors mean there. It seems like "integers modulo the polynomial $x - a$", but I don't know this structure.
Then, please, if you know what the authors are saying, explain it to me.
Thank you very much.
 A: 
It seems like "integers modulo the polynomial $x - a$", but I don't know this structure.

"Integers modulo a polynomial" doesn't make sense: the ideal of a quotient ring must be an ideal of the ring you are working in.
The ring $\frac{\mathbb{Z}[x]}{(x^n + 1, x - a)}$, on the other hand, does not suffer that problem since the ideal generated is in $\Bbb Z[x]$. By the third isomorphism theorem this ring is isomorphic to 
$$\frac{\frac{\Bbb Z[x]}{(x-a)}}{\frac{(x^n+1,x-a)}{(x-a)}}\cong \frac{\Bbb Z}{(a^n+1)}=\Bbb Z_{a^n+1}$$
The ring $\frac{\mathbb{Z}[x]}{(x^n + 1, p)}$ is much more familiar to me. By the third isomorphism theorem it is isomorphic to $$\frac{\frac{\Bbb Z[x]}{(p)}}{\frac{(x^n+1,p)}{(p)}}\cong \frac{\Bbb Z_p[x]}{(x^n+1)}$$
I thought I had heard them in the context of "skew-cyclic" or "pseudo-cyclic" or "quasi-cyclic" linear codes in algebraic coding theory, but I can't seem to track down which adjective was used or if it was widespread. 
The ordinary cyclic codes over finite fields are, of course, just ideals of the ring $\frac{F[x]}{(x^n-1)}$ where $F$ is a finite field.
