TITLE: infinite ring with non-zero finite characteristic
QUESTION [5 upvotes]: For every natural number n, i have to give a finite and a infinite ring with characteristic n. 
For the finite case is it simple: for all n $\in\mathbb{N}$, the ring $\mathbb{Z}_n$ is a finite ring with characteristic n. 
But in the infinite case i can't think of a ring... 
The only infinite ring with a finite characteristic that is not 0, is the ring of all subsets of a infinite set X. With the operations of symmetric difference and intersection. Here is the characteristic equal to 2.

REPLY [19 votes]: Consider the ring $\Bbb Z_n[x]$ of polynomials in one variable $x$ with coefficients in $\Bbb Z_n$, where $n$ is prime.  It is an infinite ring since $x^m \in \Bbb Z_n[x]$ for all positive integers $m$, and $x^{m_1} \ne x^{m_2}$ for $m_1 \ne m_2$.  But the charactetistic of $\Bbb Z_n[z]$ is clearly $n$.

REPLY [4 votes]: $A_n:= \prod \mathbb Z/n\mathbb Z$ (infinite product, fixed $n$). This is an infinite ring with characteristic $n.$