In mathematics, an ideal of a commutative ring is said to be irreducible if it cannot be written as an intersection of ideals properly contained in it.
Every prime ideal is irreducible. For noetherian rings, irreducible ideals coincide with primary ideals.
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.
See also: Laskerian ring, irreducible module.
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