Irreducibil

Irreducibil (often encountered in English texts as irreducible) is a mathematical term used to describe objects that cannot be decomposed into simpler components within a specified structure. The core idea is indivisibility relative to the governing notion of factorization or decomposition.

In algebra, irreducible elements are those that cannot be factored into a product of two nonunit elements. More precisely, in a ring with unity, a nonunit element r is irreducible if every factorization r = ab implies that either a or b is a unit. In integral domains, prime elements are irreducible, and in many familiar rings (such as unique factorization domains) every irreducible is prime, though this equivalence can fail in more general rings.

In polynomial rings, a nonconstant polynomial over a field is irreducible if it cannot be written as a product of polynomials of smaller positive degree with coefficients in that field. Irreducible polynomials play a crucial role in constructing field extensions via quotient by a maximal ideal or by modding out by an irreducible polynomial.

In algebraic geometry, an algebraic set is irreducible if it cannot be expressed as the union of two proper closed subsets. Equivalently, its vanishing ideal is prime, and irreducible varieties correspond to prime ideals in the coordinate ring.

The notion extends to topology and representation theory as well: a topological space is irreducible if it cannot be expressed as the union of two proper closed subsets, and a representation is irreducible when it has no nontrivial invariant subspaces. The exact definition of irreducibil depends on the mathematical context, but all variants share the theme of indivisibility within the given structure.