Irreducible ideal explained

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

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 Z and 3 Z contained in Z . The intersection is 6 Z , and 6 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

Z for the ideal 4 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 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

Notes and References

1. .
2. .
3. Book: Dummit. David S.. Foote. Richard M.. Abstract Algebra. 2004. John Wiley & Sons, Inc.. Hoboken, NJ. 0-471-43334-9. 683–685. Third.
4. . Theorem 1, p. 3.