Irreducible ideal

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

单词	Irreducible ideal
释义	Irreducible ideal 英语百科 Irreducible ideal In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two 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.
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