Boolean rings and Boolean algebras

[quasar987]

My professor wrote that we get a Boolean algebra from a Boolean ring (R,+,-,.,0,1) by setting xANDy=xy, xORy=x+y+xy and xNOT=1+x.

But it seems to me that xNOT is not an involution. I.e., (xNOT)NOT = 1+(1+x), which is not x.

(xNOT=-x would do the trick though)

 

[Coin]

It seems to me that

1+(1+x) = x

For several reasons. What else would it be equal to? Doesn't 1 + 1 = 0?

 

[quasar987]

I forgot about that. In an idempotent ring, x+x=0.