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

Question

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.

Answer 1

Not necessarily. Consider, in $\mathbb{Z}/4\mathbb{Z}$, the subring $R=\{\bar 0, \bar 2\}$. Then it has characteristic $2$, but as a ring, it does not contain $\mathbb{F}_2$, since it has only two elements but no identity element.

Answer 2

Another example would be the ideal $A=(X)$ in $F_p[X]$. It's a rng of characteristic $p$, but it can't contain $F_p$ since the only idempotent element is $0$.