Assume $R$ is a ring with unity such that for all $x,y \in R: xy^2=xy$, then show that $R$ is commutative ring.
I was able to show that if $(xy)^2=(yx)^2$ then the ring is a commutative ring, or the case where $(xy)^n=x^ny^n$.
Or the case where $(S; ·)$ is a semigroup which satisfies the identities $x^3 = x$ and
$x^2y = yx^2$ for all $x, y ∈ S$, then $S$ is commutative.
But I don't know how to show that for all $x,y \in R: xy^2=xy$, then show that $R$ is commutative ring,moreover the ring is a ring with unity which is different from the other questions.
Best Answer
By setting $x = 1$ and $y=-1$, one has $x = -x$ and $y^2 = y$ for all $x, y \in R$.
Let $a,b \in R$. We want to show $ab = ba$. Since $(a+b)^2 = (a+b)$, one as $ab + ba = 0$. Equivalently, $ab = -ba$. Since $x = -x$ for all $x \in R$, $-ba = ba$.