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.^[1]

Examples[edit]

• Every prime ideal is irreducible.^[2] Let ${\displaystyle J}$ and ${\displaystyle K}$ be ideals of a commutative ring ${\displaystyle R}$, with neither one contained in the other. Then there exist ${\displaystyle a\in J\setminus K}$ and ${\displaystyle b\in K\setminus J}$, where neither is in ${\displaystyle J\cap K}$ but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals ${\displaystyle 2\mathbb {Z} }$ and ${\displaystyle 3\mathbb {Z} }$ contained in ${\displaystyle \mathbb {Z} }$. The intersection is ${\displaystyle 6\mathbb {Z} }$, and ${\displaystyle 6\mathbb {Z} }$ is not a prime ideal.
• Every irreducible ideal of a Noetherian ring is a primary ideal,^[1] and consequently for Noetherian rings an irreducible decomposition is a primary decomposition.^[3]
• Every primary ideal of a principal ideal domain is an irreducible ideal.
• Every irreducible ideal is primal.^[4]

Properties[edit]

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 ${\displaystyle \mathbb {Z} }$ for the ideal ${\displaystyle 4\mathbb {Z} }$ since it is not the intersection of two strictly greater ideals.

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

See also[edit]

• Irreducible module
• Irreducible space
• Laskerian ring

References[edit]

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