Give an example of a ring with characteristic 23 but is not a field

Question

I know that Z mod 23 would have characteristic 23, but this is a field since 23 is prime. How can I go about solving this? Any help is appreciated, thanks!

Answer 1

Ring of polynomes (Z/23Z)[x^2 - 2*x + 1] as example

x-1 is a zero divizor

Answer 2

Hint: consider $\Bbb Z_{23}\times \Bbb Z_{23}$.

You know there is a certain multiplication that makes this a field. Perhaps you could find some other multiplication operation that is not quite so nice?