While reading a paper I came up with the following quotient ring definition but I am not sure what it does exactly mean:
$R=Z[x]/(x^{n}-1)$ ---> I know this one, since it has one quotient parameter which is $x^{n}-1$
$R/3=Z[x]/(3,x^{n}-1)$---> But this one has more than one qutient parameter, one is 3 and the other one $x^{n}-1$. What does this exactly mean?
Does it mean that we take the modulo of the elements of $Z[x]$ according to $x^{n}-1$ and then take the modulo again according to 3? If so, does that mean $Z[x]/(3,x^{n}-1)=Z_3[x]/(x^{n}-1)$?
In other words do we take modulo $x^{n}-1$ of the elements of $Z_3[x]$?
Edit: Does it make any difference if i change the polynomial $x^{n}-1$ to $x^{n-1}+ x^{n-2}+...+1$ or to something else?
The question you have doesn't seem to be "What is $\mathbb{Z}[x]/(3, x^{n-1})?$" so much as it seems to be "What is $(3, x^{n-1})$? To put it shortly, it is the smallest ideal of $\mathbb{Z}[x]$ that contains $3$ and $x^{n-1}$.
To put it a little bit more clearly, it is the set of all polynomials $p(x)$ such that $p(x) = 3q(x) + x^{n-1}r(x)$. To put it even more explicitly, it's the set of all polynomials where the first $x^{n - 2}$ terms have a coefficient divisible by $3$.
Put together, and skipping a lot of machinery in the background, you are right. $\mathbb{Z}[x]/(3, x^{n-1}) \simeq \mathbb{Z}_3[x]/(x^{n - 1})$.