Editing Ideal (ring theory) (section)

== 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 ring]]s. Different types of ideals are studied because they can be used to construct different types of factor rings.
* '''[[Maximal ideal]]''': A proper ideal {{mvar|I}} is called a '''maximal ideal''' if there exists no other proper ideal {{math|''J''}} with {{mvar|I}} a proper subset of {{math|''J''}}.  The factor ring of a maximal ideal is a [[simple ring]] in general and is a [[field (mathematics)|field]] for commutative rings.<ref>Because simple commutative rings are fields. See {{cite book|author=Lam|year=2001|title=A First Course in Noncommutative Rings|url={{Google books|plainurl=y|id=f15FyZuZ3-4C|page=39|text=simple commutative rings}}|page=39}}</ref>
* '''[[Minimal ideal]]''': A nonzero ideal is called minimal if it contains no other nonzero ideal.
* '''Zero ideal''': the ideal <math>\{0\}</math>.<ref>{{cite web|url=https://mathworld.wolfram.com/ZeroIdeal.html|title=Zero ideal|website=Math World|date=22 Aug 2024}}</ref>
* '''Unit ideal''': the whole ring (being the ideal generated by <math>1</math>).{{sfnp|Dummit|Foote|2004|p=243}}
* '''[[Prime ideal]]''': A proper ideal <math>I</math> is called a '''prime ideal''' if for any <math>a</math> and <math>b</math> in {{tmath|1= R }}, if <math>ab</math> is in {{tmath|1= I }}, then at least one of <math>a</math> and <math>b</math> is in {{tmath|1= I }}. The factor ring of a prime ideal is a [[prime ring]] in general and is an [[integral domain]] for commutative rings.{{sfnp|Dummit|Foote|2004|p=255}}
* '''[[Radical of an ideal|Radical ideal]]''' or [[semiprime ideal]]: A proper ideal {{mvar|I}} is called '''radical''' or '''semiprime''' if for any {{math|''a''}} in <math>R</math>, if {{math|''a''<sup>''n''</sup>}} is in {{mvar|I}} for some {{math|''n''}}, then {{math|''a''}} is in {{mvar|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 {{mvar|I}} is called a '''primary ideal''' if for all {{math|''a''}} and {{math|''b''}} in {{math|''R''}}, if {{math|''ab''}} is in {{mvar|I}}, then at least one of {{math|''a''}} and {{math|''b''<sup>''n''</sup>}} is in {{mvar|I}} for some [[natural number]] {{math|''n''}}. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
* '''[[Principal ideal]]''': An ideal generated by ''one'' element.{{sfnp|Dummit|Foote|2004|p=251}}
* {{anchor|Finitely generated ideal}}'''Finitely generated ideal''': This type of ideal is [[finitely generated module|finitely generated]] as a module.
* '''[[Primitive ideal]]''': A left primitive ideal is the [[Annihilator (ring theory)|annihilator]] of a [[simple module|simple]] left [[module (mathematics)|module]].
* '''[[Irreducible ideal]]''': An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.
* '''Comaximal ideals''': Two ideals {{mvar|I}}, {{mvar|J}} are said to be '''comaximal''' if <math>x + y = 1</math> for some <math>x \in I</math> and {{tmath|1= y \in J }}.
* '''[[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.
* '''[[Nilpotent ideal]]''':  Some power of it is zero.
* '''[[Parameter ideal]]''': an ideal generated by a [[system of parameters]].
* '''[[Perfect ideal]]''': A proper ideal {{mvar|I}} in a Noetherian ring <math>R</math> is called a '''perfect ideal''' if its [[Grade (ring theory)|grade]] equals the [[projective dimension]] of the associated quotient ring,<ref name=Matsumura1>{{cite book |last=Matsumura |first=Hideyuki |author-link=Hideyuki Matsumura |date=1987 |title=Commutative Ring Theory |url=https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1 |location=Cambridge |publisher=Cambridge University Press | page=132 |isbn=9781139171762}}</ref> {{tmath|1= \textrm{grade}(I)=\textrm{proj}\dim(R/I) }}. A perfect ideal is [[Unmixed ideal|unmixed]].
* '''[[Cohen–Macaulay_ring#unmixed|Unmixed ideal]]''': A proper ideal {{mvar|I}} in a Noetherian ring <math>R</math> is called an '''unmixed ideal''' (in height) if the height of {{mvar|I}} is equal to the height of every [[associated prime]] {{math|''P''}} of <math>R/I</math>. (This is stronger than saying that <math>R/I</math> is [[equidimensionality|equidimensional]]. See also [[Minimal_prime_ideal#Equidimensional_ring|equidimensional ring]].

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 <math>R</math> is a commutative domain with [[quotient field]] <math>K</math>. Despite their names, fractional ideals are <math>R</math> submodules of <math>K</math> with a special property. If the fractional ideal is contained entirely in <math>R</math>, then it is truly an ideal of <math>R</math>.
* '''[[Invertible ideal]]''': Usually an invertible ideal {{math|''A''}} is defined as a fractional ideal for which there is another fractional ideal {{math|''B''}} such that {{math|1=''AB'' = ''BA'' = ''R''}}. Some authors may also apply "invertible ideal" to ordinary ring ideals {{math|''A''}} and {{math|''B''}} with {{math|1=''AB'' = ''BA'' = ''R''}} in rings other than domains.