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

• 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

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

• Irreducible module
• Irreducible space
• Laskerian ring

References

1. Miyanishi, Masayoshi (1998), Algebraic Geometry, Translations of mathematical monographs, vol. 136, American Mathematical Society, p. 13, ISBN 9780821887707.
2. Knapp, Anthony W. (2007), Advanced Algebra, Cornerstones, Springer, p. 446, ISBN 9780817645229.
3. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (Third ed.). Hoboken, NJ: John Wiley & Sons, Inc. pp. 683–685. ISBN 0-471-43334-9.
4. Fuchs, Ladislas (1950), "On primal ideals", Proceedings of the American Mathematical Society, 1 (1): 1–6, doi:10.2307/2032421, JSTOR 2032421, MR 0032584. Theorem 1, p. 3.