irreducible ideal

Let R be a ring. An ideal I in R is said to be if, whenever I is an intersectionMathworldPlanetmath of two ideals: I=JK, then either I=J or I=K.

Irreducible idealsMathworldPlanetmath are closely related to the notions of irreducible elementsMathworldPlanetmath in a ring. In fact, the following holds:

Proposition 1.

If D is a gcd domain, and x is an irreducible element, then I=(x) is an irreducible ideal.


If x is a unit, then I=D and we are done. So we assume that x is not a unit for the remainder of the proof.

Let I=JK and suppose aJ-I and bK-I. Then ab=xn for some n. Let c be a gcd of a and x. So


for some dD. Since x is irreducible, either c is a unit or d is. The proof now breaks down into two cases:

  • c is a unit. Let t be a lcm of a and x. Then tc is an associate of ax. But c is a unit, t and ax are associates, so that ax is a lcm of a and x. As ab=xn, both aab and xab hold, which imply that axab. Write axr=ab, where rD. Then b=xrI, which is impossible by assumptionPlanetmathPlanetmath.

  • d is a unit. So c is an associate of x. Because c divides a, we get that xa as well, or aI, which is again impossible by assumption.

Therefore, the assumption that J-I and K-I is false, which is the same as saying JI or KI. But IJ and IK, either I=J or I=K, or I is irreducible. ∎

Remark. In a commutativePlanetmathPlanetmathPlanetmath Noetherian ringMathworldPlanetmath, the notion of an irreducible ideal can be used to prove the Lasker-Noether theorem: every ideal (in a Noetherian ring) has a primary decomposition.


  • 1 D.G. Northcott, Ideal Theory, Cambridge University Press, 1953.
  • 2 H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.
  • 3 M. Reid, Undergraduate Commutative Algebra, Cambridge University Press, 1996.
Title irreducible ideal
Canonical name IrreducibleIdeal
Date of creation 2013-03-22 18:19:47
Last modified on 2013-03-22 18:19:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 13E05
Classification msc 13A15
Classification msc 16D25
Synonym indecomposable ideal
Related topic IrreducibleElement