Ideal (ring Theory) - Types of Ideals

Types of Ideals

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.

• Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings.
• Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
• Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least one of a and b is in I. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
• Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in R, if an is in I for some n, then a is in I. The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings.
• Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and bn is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
• Principal ideal: An ideal generated by one element.
• Finitely generated ideal: This type of ideal is finitely generated as a module.
• Primitive ideal: A left primitive ideal is the annihilator of a simple left module. A right primitive ideal is defined similarly. Actually (despite the name) the left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. A factor rings constructed with a right (left) primitive ideals is a right (left) primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
• Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals which properly contain it.
• Comaximal ideals: Two ideals are said to be comaximal if for some and .
• Regular ideal: This term has multiple uses. See the article for a list.
• Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent.

Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details:

• Fractional ideal: This is usually defined when R is a commutative domain with quotient field K. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.
• Invertible ideal: Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that AB=BA=R. Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with AB=BA=R in rings other than domains.