Can a ring of positive characteristic have infinite number of elements?

For curiosity: can a ring of positive characteristic ever have infinite number of distinct elements? (For example, in $\mathbb{Z}/7\mathbb{Z}$, there are really only seven elements.) We know that any field/ring of characterisitc zero must have infinite elements, but I am not sure what happens above.

Solution 1:

Consider $\mathbf{Z}/p\mathbf{Z}[x]$.

Solution 2:

Start with any infinite set $X$ and let $R$ be the set of all subsets of $X$. With the operations of symmetric difference (as addition) and intersection (as multiplication), $R$ is a ring of characteristic $2$.

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