Irreducible ( mathematics ) 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 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 . This disambiguation page lists articles associated with the same title . If an internal link led you here , you may wish to change the link to point directly to the intended article . Categories : Disambiguation | Mathematical disambiguation 