Idempotent Elements of Ring with No Proper Zero Divisors

Theorem
Let $$\left({R, +, \circ}\right)$$ be a non-null ring with no zero divisors.

Let $$x \in R$$.

Then:


 * $$x \circ x = x \iff x \in \left\{{0_R, 1_R}\right\}$$

That is, the only elements of $$R$$ that are idempotent are zero and unity.

Proof

 * We have $$0_R \circ 0_R = 0_R$$, so that sorts out one element.


 * All elements of $$R^*$$ that are not zero divisors are cancellable.

Therefore all elements of $$R^*$$ are cancellable.

Suppose $$x \circ x = x$$ where $$x \ne 0_R$$.

Then $$x \circ x = x = x \circ 1_R$$.

As $$x$$ is cancellable, the result follows.