Definition:Irreducible


 * Ring Theory:
 * Irreducible Element: An element of an integral domain $D$ is irreducible it has no non-trivial factorization in $D$.
 * Irreducible Polynomial: A polynomial is irreducible over a field it cannot be represented as the product of two or more non-constant polynomials over that field.


 * Topology:
 * Irreducible Space: A topological space is irreducible it cannot be represented as a decomposition of closed sets.


 * Representation Theory:
 * Irreducible Linear Representation: A linear representation $\rho: G \to \operatorname{GL} \left({V}\right)$ is irreducible  it has no non-trivial subspace $W$ which is invariant for every linear operator in the set $\left\{{\rho \left({g}\right): g \in G}\right\}$.
 * Irreducible G-Module: A $G$-module is irreducible its corresponding linear representation is irreducible.