Assertion
In a boolean ring every prime ideal is a maximal ideal.
(A boolean ring is a ring for which x² = x for all x in R, that is, R consists only of idempotent elements.)
Proof
Let P be a a prime ideal in a boolean ring R with 1.
Then is L = R/P a boolean ring without zero divisors.
L can’t have more than two elements:
Let be x ≠ 0 and y ≠ 0 two elements of L and z := xy.
z = xy
=> xz = xxy
=> xz = xy | da xx = x² = x
=> 0 = xy – xz
=> 0 = x(y-z)
L has no zero divisors, so we have y = z.
In the same way we conclude x = z and therefore x = y.
Therefore L has not more than one element besides 0.
In this case R/P has exactly two elements, because R ≠ P.
R/P is isomorphic to ℤ/2ℤ and therefore a field.
We have P is a maximal ideal.
q.e.d.