Irreducible ideal

In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.

Every prime ideal is irreducible. Every irreducible ideal of a Noetherian ring is a primary ideal, and consequently for Noetherian rings an irreducible decomposition is a primary decomposition. Every primary ideal of a principal ideal domain is an irreducible ideal. Every irreducible ideal is a primal ideal.

An element of an integral domain is prime if, and only if, an ideal generated by it is a nonzero prime ideal. This is not true for irreducible ideals: an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in ${\displaystyle \mathbb {Z} }$ for the ideal ${\displaystyle 4\mathbb {Z} }$: It is not the intersection of two strictly greater ideals.

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