Idempotent Elements of Ring with No 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.

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

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

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$: Theorem $21.3$