Idempotent element

In abstract algebra, an element x of a set with a binary operation ∗ is called an idempotent element (or just an idempotent) if x ∗ x = x. This reflects the idempotence of the binary operation on that particular element.

Idempotents are especially prominent in ring theory. For general rings, elements idempotent under multiplication are tied with decompositions of modules, as well as to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.