# Irreducible

From Example Problems

In mathematics, the term * irreducible* is used in several ways.

- In abstract algebra,
**irreducible**can be an abbreviation for irreducible element; for example an irreducible polynomial.

- In commutative algebra, a commutative ring
*R*is**irreducible**if its prime spectrum, that is, the topological space Spec*R*, is an irreducible topological space.

- A directed graph is
**irreducible**if, given any two vertices, there exists a path from the first vertex to the second. A digraph is irreducible iff its adjacency matrix is irreducible.

- In the theory of manifolds, an
*n*-manifold is**irreducible**if any embedded (*n*−1)-sphere bounds an embedded*n*-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.The notions of irreducibility in algebra and manifold theory are related. An

*n*-manifold is called prime, if it cannot be written as a connected sum of two*n*-manifolds (neither of which is an*n*-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over*S*^{1}and the twisted 2-sphere bundle over*S*^{1}.

- A matrix is
**irreducible**if it cannot be made block upper triangular via a matrix permutation.

- In representation theory, an
**irreducible representation**is a nontrivial representation with no nontrivial subrepresentations. Similarly, an**irreducible module**is another name for a simple module.

- A topological space is
**irreducible**if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.

- In universal algebra,
**irreducible**can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.