Answer to Question #350986 in Abstract Algebra for Deq

Question

3.3. If in a ring R every x ∈ R satisfies x² = x, prove that R

must be commutative

(A ring in which x² = x for all elements is called a Boolean ring).

Answer 1

Let $x,y\in R$. Then $(x + y)^2 = (x + y)(x + y) = x^2 + xy + yx + y^2$

Since $x^2 = x$ and $y^2 = y$ we have $x + y = x + xy + yx + y$.

Hence $xy = −yx$.

But for every $x\in R$: $(−x) = (−x)^2 = (−x)(−x) = x^2 = x$.

Hence $−yx = yx$ i.e. we obtain $xy = yx$.