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 $$2 \mathbb Z$$ and $$3 \mathbb Z$$ contained in $$\mathbb Z$$. The intersection is $$6 \mathbb Z$$, and $$6 \mathbb Z$$ 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.

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 $$\mathbb Z$$ for the ideal $$4 \mathbb Z$$ 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 $$\operatorname{Spec} R$$. 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 $$(x^2,xy,y^2) = (x^2,y) \cap (x,y^2) $$.