irreducible ideal
Let $R$ be a ring. An ideal $I$ in $R$ is said to be if, whenever $I$ is an intersection^{} of two ideals: $I=J\cap K$, then either $I=J$ or $I=K$.
Irreducible ideals^{} are closely related to the notions of irreducible elements^{} in a ring. In fact, the following holds:
Proposition 1.
If $D$ is a gcd domain, and $x$ is an irreducible element, then $I\mathrm{=}\mathrm{(}x\mathrm{)}$ is an irreducible ideal.
Proof.
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 $I=J\cap K$ and suppose $a\in JI$ and $b\in KI$. Then $ab={x}^{n}$ for some $n\in \mathbb{N}$. Let $c$ be a gcd of $a$ and $x$. So
$$cd=x$$ 
for some $d\in D$. 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 $ab={x}^{n}$, both $a\mid ab$ and $x\mid ab$ hold, which imply that $ax\mid ab$. Write $axr=ab$, where $r\in D$. Then $b=xr\in I$, which is impossible by assumption^{}.

•
$d$ is a unit. So $c$ is an associate of $x$. Because $c$ divides $a$, we get that $x\mid a$ as well, or $a\in I$, which is again impossible by assumption.
Therefore, the assumption that $JI\ne \mathrm{\varnothing}$ and $KI\ne \mathrm{\varnothing}$ is false, which is the same as saying $J\subseteq I$ or $K\subseteq I$. But $I\subseteq J$ and $I\subseteq K$, either $I=J$ or $I=K$, or $I$ is irreducible. ∎
Remark. In a commutative^{} Noetherian ring^{}, the notion of an irreducible ideal can be used to prove the LaskerNoether theorem: every ideal (in a Noetherian ring) has a primary decomposition.
References
 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  20130322 18:19:47 
Last modified on  20130322 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 