Every commutative ring of characteristic $p$ contains $\mathbb F_p$ as a subring?

I know that if a commutative ring with unity is of characteristic $p$ then it will contain $\mathbb F_p$ as a subring, but if the ring is commutative with characteristic $p$ and without unity then is it possible to find such a map between the ring and $\mathbb Z$ such that we can show that the ring contains $\mathbb F_p$ as a subring?

Thank you.