Irreducibiliteet

Irreducibiliteet (irreducibility) is a fundamental notion describing objects that cannot be decomposed into simpler factors under a given type of factorization. The idea appears in several branches of mathematics, often with related but context-dependent definitions.

In ring theory, an element a of a commutative ring with unity is irreducible if a is nonzero, not a unit, and whenever a = bc, then either b or c is a unit. In integral domains, prime elements are irreducible; in general domains irreducible does not imply prime. In principal ideal domains and unique factorization domains, irreducible and prime coincide.

In polynomial rings over a field F, a nonconstant polynomial f is irreducible if it cannot be factored as f = gh with polynomials g, h of positive degree in F[x]. Over larger rings, irreducibility may depend on the coefficient ring.

In algebraic geometry, an algebraic set V is irreducible if it cannot be written as a union of two proper closed subsets. Equivalently, the vanishing ideal I(V) is a prime ideal; over algebraically closed fields, this links to coordinate rings being integral domains. Irreducible varieties correspond to indivisible geometric pieces and decompose into a finite union of irreducible components.

In topology, a nonempty space X is irreducible if it cannot be expressed as the union of two proper closed subsets, or equivalently every two nonempty open subsets intersect.

The concept also extends to modules, schemes, and other structures. The term appears in Finnish as irreduibiliteetti or irreduibiliteet, used similarly in mathematical contexts.