Example of rings of the same positive characteristic that do not embed into their tensor product?

Example of rings of the same positive characteristic that do not embed
into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got
me wondering:
Give an example of commutative rings $A$ and $B$ with
$\operatorname{char}A=\operatorname{char}B$ such that the map $$A
\longrightarrow\ A\otimes_{\Bbb{Z}}B:\ a\ \longmapsto\ a\otimes1,$$ is not
injective.
Examples with $\operatorname{char}A\neq\operatorname{char}B$ are of course
abundant, and examples with $\operatorname{char}A=\operatorname{char}B=0$
aren't too difficult either. But I am unable to find an example with
$\operatorname{char}A=\operatorname{char}B>0$. An example would be very
welcome, but a clue as to where to look for one would also be much (or
even more?) appreciated.