Idempotent Elements of Ring with No Proper Zero Divisors

Theorem

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

Let $x \in R$.

Then:

$x \circ x = x \iff x \in \set {0_R, 1_R}$

That is, the only elements of $\struct {R, \circ}$ 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: $R^* = R \setminus \set {0_R}$.

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

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Theorem $21.3$