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 iff its Adjacency Matrix is irreducible.
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 , 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.
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 .
