nonreducible

Nonreducible is an adjective meaning not capable of being reduced or simplified. In mathematics and related fields, the term is often used synonymously with irreducible, though nonreducible is less common in modern practice. The exact meaning depends on the object being considered and the notion of “reduction” appropriate to that context.

In algebra, an element of a ring is nonreducible (irreducible) if it is not a unit and whenever it equals a product ab, at least one of a or b is a unit. Over a field, a nonconstant polynomial is irreducible if it cannot be written as a product of two nonconstant polynomials with coefficients in that field. The concept generalizes to objects like ideals and modules, where irreducibility concerns inability to express the object as a nontrivial combination of smaller, simpler pieces.

In topology and algebraic geometry, an irreducible space or variety cannot be expressed as the union of two proper closed subsets. In this setting irreducibility reflects an inseparability of the space’s structure under the Zariski topology or other related frameworks.

In representation theory, an irreducible (or nonreducible) representation has no proper nonzero invariant subspaces. A related notion is indecomposability, where a representation cannot be written as a direct sum of two nonzero subrepresentations; in general, irreducibility implies indecomposability.

Usage notes: many mathematical writers prefer irreducible as the standard term; nonreducible appears mainly in older texts or in non-English contexts. See also irreducible, reducible, prime, and indecomposable.