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| | Hi there. Let me begin by introducing the writer, her name is Myrtle Cleary. North Dakota is where me and my spouse reside. To gather cash is what her family members and her enjoy. For many years I've been working as a payroll clerk.<br><br>Have a look at my blog; [http://service.mobile.rotanaradio.jo/node/53196 std testing at home] |
| In [[mathematics]], the concept of '''irreducibility''' is used in several ways.
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| * 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]].
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| * [[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]]''.
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| * 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]].
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| * 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]].)
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| * 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.
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| * In the theory of [[manifold]]s, 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 (mathematics)|category]], such as the category of differentiable manifolds or the category of piecewise-linear manifolds.
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| 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)]].
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| * 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]].
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| * 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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| * A [[3-manifold]] is [[P²-irreducible]] if it is irreducible and contains no [[2-sided]] <math>\mathbb RP^2</math> ([[real projective plane]]).
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| * 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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| {{DEFAULTSORT:Irreducibility (Mathematics)}}
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| [[Category:Mathematical terminology]]
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Hi there. Let me begin by introducing the writer, her name is Myrtle Cleary. North Dakota is where me and my spouse reside. To gather cash is what her family members and her enjoy. For many years I've been working as a payroll clerk.
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