Idempotents

An idempotent is an element e in an algebraic structure with a binary operation such that e squared equals e: e^2 = e. In rings or monoids this means e·e = e. For a function f: X → X, idempotence means f(f(x)) = f(x) for all x in X.

Basic consequences include e(1 − e) = 0 in rings with identity. If e is central (commutes with all elements), the ring R can be decomposed as a direct product R ≅ R e × R(1 − e). Conversely, a nontrivial central idempotent corresponds to a nontrivial decomposition of the ring; rings without such idempotents are called indecomposable.

Examples illustrate the concept. The ring of integers Z has only the trivial idempotents 0 and 1. The matrix ring M_n(F) has many idempotents; projection matrices like diag(1, ..., 1, 0, ..., 0) are idempotent. In linear algebra, projection operators P with P^2 = P are idempotent, and a vector space V decomposes as im P ⊕ ker P, with P acting as the identity on im P and as zero on ker P.

In commutative algebra, idempotents correspond to decompositions of Spec: a ring decomposes as a product of rings precisely when it has a nontrivial idempotent, reflecting a disconnection in its spectrum. In category theory, idempotent morphisms (morphisms satisfying e ∘ e = e) are studied via the process of splitting, leading to constructions like the Karoubi envelope that formally adjoin splittings for all idempotents.