infinite ring with non-zero finite characteristic

Question

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.

Answer 1

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 $n$.

Answer 2

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