Irreducible ideal

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

Examples

• Every prime ideal is irreducible. Let $J$ and $K$ be ideals of a commutative ring $R$, with neither one contained in the other. Then there exist $a\in J\setminus K$ and $b\in K\setminus J$, where neither is in $J\cap K$ but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals $2\mathbb{\mathbb{Z}}$ and $3\mathbb{\mathbb{Z}}$ contained in $\mathbb{\mathbb{Z}}$. The intersection is $6\mathbb{\mathbb{Z}}$, and $6\mathbb{\mathbb{Z}}$ is not a prime ideal.
• 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 primal.
• Every irreducible ideal that is also radical is prime. The ideal $4\mathbb{\mathbb{Z}}$ is an example of an irreducible ideal in $\mathbb{\mathbb{Z}}$ that is not radical and not a prime ideal.

Properties

An element of an integral domain is prime if and only if the ideal generated by it is a non-zero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in $\mathbb{\mathbb{Z}}$ for the ideal $4\mathbb{\mathbb{Z}}$ since it is not the intersection of two strictly greater ideals.

In algebraic geometry, if an ideal $I$ of a ring $R$ is irreducible, then $V(I)$ is an irreducible subset in the Zariski topology on the spectrum $\mathrm{Spec}R$. The converse does not hold; for example the ideal $(x^2,xy,y^2)$ in $\mathbb{ℂ}[x,y]$ defines the irreducible variety consisting of just the origin, but it is not an irreducible ideal as $(x^2,xy,y^2)=(x^2,y)\cap (x,y^2)$.

See also

• Irreducible module
• Irreducible space
• Laskerian ring