Let $(R,m)$ be a local ring of Krull dimension $0$ and such that $R/m$ is a finite field. Then is it true that $R$ has positive characteristic ?
If we assume $R$ is Noetherian then I know $R$ becomes finite and then it is trivial. I don't know what happens if $R$ is not Noetherian.
It is elementary. Suppose $p$ is the characteristic of the residue field. Then $p\cdot 1\in J(R)$, which is a nil ideal when $R$ is $0$-dimensional.
Therefore there exists a $k$ such that $p^k\cdot 1=0$. This shows $R$ has positive characteristic.