Idempotent element

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

Idempotents are the algebraic shadows of direct product decompositions: under suitable hypotheses, they produce ring splittings as in idempotent product decompositions . In many commutative settings, nontrivial idempotents correspond to disconnectedness of $\mathrm{Spec}(R)$.

Examples:

• In any unital ring, $0$ and $1$ are idempotent.
• In $M_2(k)$, the matrix $\begin{pmatrix}1&0\\0&0\end{pmatrix}$ is idempotent.
• In an integral domain, every idempotent is $0$ or $1$.