Idempotent Elements of Ring with No Proper Zero Divisors

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Theorem

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


Let $R^*$ be the ring $R$ without the zero: $R^* = R \setminus \left\{{0_R}\right\}$.

By Ring Element is Zero Divisor iff not Cancellable, 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.

$\blacksquare$


Sources