Editing Irreducibility (mathematics)

In [[mathematics]], the concept of '''irreducibility''' is used in several ways.

* A [[polynomial]] over a [[field (mathematics)|field]] may be an [[irreducible polynomial]] if it cannot be factored over that field.
* In [[abstract algebra]], '''irreducible''' can be an abbreviation for [[irreducible element]] of an [[integral domain]]; for example an [[irreducible polynomial]].
*  In [[representation theory]], an '''[[irreducible representation]]''' is a nontrivial [[representation theory|representation]] with no nontrivial proper subrepresentations. Similarly, an '''irreducible module''' is another name for a [[simple module]].
* [[Absolutely irreducible]] is a term applied to mean [[irreducible]], even after any [[finite extension]] of the [[field (mathematics)|field]] of coefficients.  It applies in various situations, for example to irreducibility of a [[linear representation]], or of an [[algebraic variety]]; where it means just the same as ''irreducible over an [[algebraic closure]]''.
* 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 [[matrix (mathematics)|matrix]] is irreducible if it is not [[similar matrix|similar]] via a [[permutation matrix|permutation]] to a [[block matrix|block]] [[upper triangular matrix]] (that has more than one block of positive size).  (Replacing non-zero entries in the matrix by one, and viewing the matrix as the [[adjacency matrix]] of a [[directed graph]], the matrix is irreducible if and only if such directed graph is [[Connectivity (graph theory)|strongly connected]].) A detailed definition is [[Perron–Frobenius theorem#Classification of matrices|given here]].
* Also, a [[Markov chain]] is [[Markov chain#Properties|irreducible]] if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.
* In the theory of [[manifold]]s, an ''n''-manifold is [[irreducible manifold|irreducible]] if any embedded (''n''&nbsp;−&nbsp;1)-sphere bounds an embedded ''n''-ball.  Implicit in this definition is the use of a suitable [[category (mathematics)|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 [[Connected sum|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''<sup>1</sup> and the twisted 2-sphere bundle over ''S''<sup>1</sup>.  See, for example, [[Prime decomposition (3-manifold)]].
* A [[topological space]] is [[irreducible space|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 space]]s. 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]].
* A [[3-manifold]] is [[P²-irreducible]] if it is irreducible and contains no [[2-sided]] <math>\mathbb RP^2</math> ([[real projective plane]]).
* An [[irreducible fraction]] (or fraction in lowest terms) is a [[vulgar fraction]] in which the [[numerator]] and [[denominator]] are smaller than those in any other equivalent fraction.

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