Thread: Boolean commutative ring

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?

Originally Posted by DanielThrice

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?

These questions seem to be coming from overarching question, did you forget to include it?

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.

By F2 do you mean or just some generic field?

Thank you, that's a pretty clear way of looking at it

Originally Posted by Drexel28

These questions seem to be coming from overarching question, did you forget to include it?

And no this is a basic theory kinda problem, no specifics

F2 is the finite field of 2 elements. Z2 is another name for it, yes.