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Every Boolean ring is commutative

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.15

A ring $R$ is called Boolean if $a^2 = a$ for all $a \in R$. Prove that every Boolean ring is commutative.

Solution: Note first that for all $a \in R$, $$-a = (-a)^2 = (-1)^2 a^2 = a^2 = a.$$ Now if $a,b \in R$, we have $$a + b = (a+b)^2 = a^2 + ab + ba + b^2 = a + ab + ba + b.$$ Thus $ab + ba = 0$, and we have $ab = -ba$. But then $ab = ba$. Thus $R$ is commutative.


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