Irreducible

Irreducible is a term used across mathematics to denote an object that cannot be decomposed into simpler constituents within a given structure. The precise meaning depends on the context, with common usages in algebra, geometry, and representation theory.

In ring theory, an element p of a commutative ring with unity is irreducible if p is not a unit and whenever p = ab, then either a or b is a unit. Irreducibles are the basic building blocks for factorization; in a unique factorization domain every irreducible is prime, but not every irreducible is prime in general rings.

In polynomial algebra, a polynomial over a field is irreducible if it cannot be factored into polynomials of smaller positive degree with coefficients in that field. For example, over the real numbers, x^2 + 1 is irreducible, while x^2 − 1 factors as (x − 1)(x + 1).

In algebraic geometry, an algebraic set is irreducible if it cannot be expressed as the union of two proper closed subsets; equivalently, its defining ideal is prime.

In representation theory, an irreducible (or simple) representation has no nontrivial invariant subspaces under the action of the group or algebra. If a representation contains such a subspace, it is reducible and may decompose into a direct sum of smaller representations in suitable categories.

The overarching idea in all these contexts is that irreducible objects resist nontrivial decomposition; they serve as the atomic constituents from which more complex structures are built.