Homomorphism between rings

Question

What is a homomorphism between the following rings:

The collection P(Z) of subsets of $Z$, with addition $A ⊕ B := A∆B$ and multiplication $A◦ B := A ∩ B$, and the integers mod 2.

Answer 1

Hint: For any set $Z$, the ring homomorphisms $P(Z)\to \Bbb Z/2\Bbb Z$ correspond to the ultrafilters of $Z$.

For a specific example, let $z\in Z$ arbitrary, and send $S\mapsto 0$ if $z\notin S$ and $S\mapsto 1$ if $z\in S$.