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 $a\in J-I$ and $b\in K-I$. 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 $J-I\ne \mathrm{\varnothing }$ and $K-I\ne \mathrm{\varnothing }$ 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 Lasker-Noether theorem: every ideal (in a Noetherian ring) has a primary decomposition.

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