abstract algebra – Boolean ring prime ideals

I’m asked 2 question:

1- Prove or disprove: In every Boolean ring prime ideals are maximal.

2- Prove or disprove: If $R$ is Boolean and $p$ is a prime ideal of $R$, then the ring homomorphism $Rto R_p$ is always finite.

try for 1:
$bar R=R/p$ is a boolian domain. boolian domain is a field with 2 elements. (if $x in bar R$, then x(1-x)=0 so x=0 or x=1 ). therefore R/p is field and $p$ is maximal. is this proof true?

—- for 2, i dont know i should prove or disprove. i need help and clue please.