Algebra, Problems For Fun (36)

__Definition__: An element $\displaystyle r$ in a ring $\displaystyle R$ is called central if $\displaystyle r$ is in the center of $\displaystyle R,$ i.e. $\displaystyle rs=sr,$ for all $\displaystyle s \in R.$

1) Suppose $\displaystyle R$ is a ring in which $\displaystyle x^2=0 \Longrightarrow x=0.$ Let $\displaystyle e \in R$ be an idempotent, i.e. $\displaystyle e^2=e.$ Prove that $\displaystyle e$ is central.

__Hint__:

2) Use 1) to give a short proof of this very special case of Jacobson's theorem: if $\displaystyle x^3=x,$ for all $\displaystyle x$ in a ring with identity $\displaystyle R,$ then $\displaystyle R$ is commutative.

__Hint__:

3) This time suppose $\displaystyle x^4=x,$ for all $\displaystyle x$ in a ring with identity $\displaystyle R.$ Use 1) to prove that $\displaystyle R$ is commutative.

__Hint__:

4) __Challenge__: Suppose $\displaystyle x^5=x,$ for all $\displaystyle x$ in a ring with identity $\displaystyle R.$ Use 1) to prove that $\displaystyle R$ is commutative.