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

Question

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 characteristic zero must have infinite elements, but I am not sure what happens above.

Answer 1

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

Answer 2

Besides @Brandon’s most economical example, any field (such as $\mathbb Z/p\mathbb Z$) has an algebraic closure, and an algebraically closed field can not be finite.

Answer 3

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 $2$.

Answer 4

Let $p$ be a prime, $l > 1$. The nested union $\cup_{i \in \mathbb{N}} K_i$ is yet another example; where $K_i$ is the unique field extension of $\mathbb{F}_p$ with $|K : F_p| = l^i$.