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


* In [[abstract algebra]], '''irreducible''' can be an abbreviation for [[irreducible element]] of an [[integral domain]]; for example an [[irreducible polynomial]].  
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*  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]].)
 
* Also, a [[Markov chain]] is '''[[Markov chain#Reducibility|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''' if any embedded (''n''&nbsp;&minus;&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.
 
{{DEFAULTSORT:Irreducibility (Mathematics)}}
[[Category:Mathematical terminology]]

Revision as of 00:48, 12 February 2014

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