The question is motivated by the following multiple-choice problem:
If $R$ is a ring with the property that $r=r^2$ for each $r\in R$, which of the following must be true?
I. $r+r=0$ for each $r\in R$.
II. $(r+t)^2=r^2+t^2$ for each $r,t\in R$.
III. $R$ is commutative.
Here are my questions:
- What theorems do I need to solve the problem above?
- Why is a ring with the property that $r=r^2$ for each $r\in R$ so special? Is there a name for such rings?