What is $\mathbb{Z_{n}}\left [ x \right ]$ 
Question: Show that $\mathbb{Z_{n}}\left [ x \right ]$ has characteristic $n$.

What does $\mathbb{Z_{n}}\left [ x \right ]$ stands for? I'm very sure this is not the gaussian ring.

Let $\mathbb{Z}_n$ be the set of integers $\{0,1,\ldots,n-1\}$ equipped with the operations of addition mod $n$ and multiplication mod $n$.  It can be shown this structure is a ring.  $\mathbb{Z}_n[x]$ is defined as the set of polynomials of the form $a_n x^n + \cdots + a_1 x + a_0$, where $a_i \in \mathbb{Z}_n$ equipped with the usual operations of addition and multiplication of polynomials.  This structure is also a ring.  
If we take any polynomial $f(x)$ in $\mathbb{Z}_n[x]$, and add $f(x)$ to itself $n$ times, we get $0$ because $a_i + a_i + \cdots + a_i = na_i = 0 $ in $\mathbb{Z}_n$.  Recall that the characteristic of a ring $R$ is defined to be the smallest value of $k$ such that $r+r+\cdots+r=rk=0$ for all $r \in R$.  It is clear then that the characteristic of $\mathbb{Z}_n[x]$ is at most $n$.  By considering the polynomial $f(x)=a_0=1$, we see that the number of times we need to add $f(x)=1$ to itself to get 0 is at least $n$.  Hence, the characteristic of $\mathbb{Z}_n[x]$ is exactly $n$.
