# Irreducibility

With **irreducibility** or as **irreducible** be called:

- Irreducible representation, a representation that is simple or indivisible , see Representation Theory (Group Theory) #Glossary
- Irreducible element, an element of a ring that cannot be written as the product of two non-units, see Ring (Algebra) # Irreducibility
- Irreducible ideal , a real ideal in a commutative ring that is not the intersection of two really larger ideals
- Irreducible Markov chain , a Markov chain in which one can get from any state to any other with a positive probability
- Irreducible matrix , a matrix that cannot be brought to an upper or lower block triangle shape using permutation matrices
- Irreducible operator , an operator on an ordered vector space that only has and as invariant ideals . Note that this is compatible with the definition of an irreducible matrix.
- Irreducible polynomial , a polynomial that cannot be written as the product of two non-invertible polynomials
- Irreducible topological space , a topological space that cannot be written as the union of two closed true subsets
- Irreducible 3-manifold , a 3-manifold that cannot be decomposed along 2-spheres
- a system that cannot be traced back to underlying units, see emergence