I'm just starting work on Boolean rings. I understand so far that if we have some commutative Boolean ring with unity, that for every we have .
I would think then that some ideal in would make the factor ring also boolean. How can I prove that this factor ring is also boolean?
As for maximals, how is it we would prove that every proper nontrivial prime ideal is maximal?
if R is boolean, then for any quotient R/I, we have: (α + I)(α + I) = α^2 + I = α + I, so R/I is boolean.
suppose I is a proper non-trivial prime ideal. then ab in I implies either a in I, or b in I, so R/I is an integral domain. but R/I is also boolean, so if a ≠ 0 in R/I,
then a = a^2 = a^3. thus a(1) = a(a^2), so a^2 = 1. hence a = 1 or -1. but -1 = (-1^2) = 1, so R/I = {0,1}.
from distributivity we have (1 + 1)(1 + 1) = 1 + 1 + 1 + 1, from the fact that R/I is boolean, we have (1 + 1)(1 + 1) = 1 + 1.
thus 1 + 1 = 0, so R/I is isomorphic to F2. since F2 is a field, I must be maximal.