Irreducible (mathematics)

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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 if and only if 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 S¹ and the twisted 2-sphere bundle over S¹.

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.

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