Irreducibility (mathematics)
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. ▪ In representation theory, an
irreducible representation is a nontrivial 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 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 is
irreducible if it is not similar via a permutation to a block upper triangular matrix.