Idempotent element

Definition

Let $R$ be a ring. An element $e \in R$ is idempotent if $e^2 = e$.

Remarks

In every ring, $0$ and $1$ are idempotent. In $\mathbb{Z}/6\mathbb{Z}$, the classes of $3$ and $4$ are idempotent, because $9 \equiv 3$ and $16 \equiv 4 \pmod 6$. If $e$ is idempotent, then $1 - e$ is idempotent, and $e (1 - e) = 0$. In the product $R \times S$, the elements $(1, 0)$ and $(0, 1)$ are idempotent.

Explanations

L1 human

An idempotent does not change when you square it. Among the integers, only $0$ and $1$ have this property. Other rings can have more idempotents, for example the projection matrices.