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A maximal element in the [[Partially ordered set|partially ordered set]] of proper ideals of a corresponding algebraic structure. Maximal ideals play an essential role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629601.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629602.png" /> regarded as a left (respectively, right) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629603.png" />-module relative to a left (respectively, right) maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629604.png" /> is irreducible (cf. [[Irreducible module|Irreducible module]]); a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629605.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629606.png" /> into the field of endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629607.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629608.png" />. The kernel of all such representations, that is, the set of elements of the ring which are mapped to zero by all representations, is called the Jacobson radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m0629609.png" />; it coincides with the intersection of all maximal left (also, all right) ideals.
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In the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296010.png" /> of continuous real-valued functions on a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296011.png" />, the set of functions vanishing at some fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296012.png" /> is a maximal ideal. Such ideals exhaust all maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296013.png" />. This relation between the points of the interval and the maximal ideals has resulted in the construction of various theories for representing rings as rings of functions on a topological space.
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The [[Zariski topology|Zariski topology]] on the set of prime ideals (cf. [[Prime ideal|Prime ideal]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296014.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296015.png" /> has weak separation properties (that is, there are non-closed points). A similar topology in the non-commutative case can be introduced on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296016.png" /> of primitive ideals (cf. [[Primitive ideal|Primitive ideal]]), which are the annihilators of irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296018.png" />-modules. The set of maximal ideals, and in the non-commutative case, of maximal primitive ideals, forms a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296019.png" /> which satisfies the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296021.png" />-separation axiom.
+
A maximal element in the [[Partially ordered set|partially ordered set]] of proper ideals of a corresponding algebraic structure. Maximal ideals play an essential role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module  $  M = R / I $
 +
of  $  R $
 +
regarded as a left (respectively, right)  $  R $-
 +
module relative to a left (respectively, right) maximal ideal  $  I $
 +
is irreducible (cf. [[Irreducible module|Irreducible module]]); a homomorphism  $  \phi $
 +
of  $  R $
 +
into the field of endomorphisms of  $  M $
 +
is a representation of  $  R $.
 +
The kernel of all such representations, that is, the set of elements of the ring which are mapped to zero by all representations, is called the Jacobson radical of  $  R $;
 +
it coincides with the intersection of all maximal left (also, all right) ideals.
 +
 
 +
In the ring  $  R = C [ a , b ] $
 +
of continuous real-valued functions on a closed interval  $  [ a , b ] $,
 +
the set of functions vanishing at some fixed point  $  x _ {0} $
 +
is a maximal ideal. Such ideals exhaust all maximal ideals of  $  R $.
 +
This relation between the points of the interval and the maximal ideals has resulted in the construction of various theories for representing rings as rings of functions on a topological space.
 +
 
 +
The [[Zariski topology|Zariski topology]] on the set of prime ideals (cf. [[Prime ideal|Prime ideal]]) $  \mathop{\rm Spec}  R $
 +
of a ring $  R $
 +
has weak separation properties (that is, there are non-closed points). A similar topology in the non-commutative case can be introduced on the set $  \mathop{\rm Spec}  R $
 +
of primitive ideals (cf. [[Primitive ideal|Primitive ideal]]), which are the annihilators of irreducible $  R $-
 +
modules. The set of maximal ideals, and in the non-commutative case, of maximal primitive ideals, forms a subspace $  \mathop{\rm Specm}  R \subset  \mathop{\rm Spec}  R $
 +
which satisfies the $  T _ {1} $-
 +
separation axiom.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Maximal ideals also play an important role in the structure and representation theory of lattices (particularly distributive lattices). In a [[Distributive lattice|distributive lattice]], as in a commutative ring, all maximal ideals are prime; the converse implication holds in a [[Boolean algebra|Boolean algebra]], and indeed a distributive lattice in which all prime ideals are maximal is necessarily Boolean. As with rings, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296022.png" /> of maximal ideals of a distributive lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296023.png" /> can be topologized as a subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296024.png" /> of all prime ideals, and it is a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296025.png" />-space; moreover, every compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296026.png" />-space arises in this way. A distributive lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296027.png" /> is said to be normal if, given elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296029.png" />, there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296030.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296032.png" />. Normal distributive lattices can be characterized as those for which every prime ideal is contained in a unique maximal ideal, or equivalently as those for which there is a continuous [[Retraction|retraction]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296033.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296034.png" />; they have the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296035.png" /> is a Hausdorff space. For a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296036.png" />, the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296037.png" /> of open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296038.png" /> is normal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296039.png" /> is a [[Normal space|normal space]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296040.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296041.png" />-space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296042.png" /> yields a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296043.png" /> [[Compactification|compactification]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296044.png" />, which coincides with the [[Stone–Čech compactification|Stone–Čech compactification]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296045.png" /> is normal (see [[Wallman compactification|Wallman compactification]]).
+
Maximal ideals also play an important role in the structure and representation theory of lattices (particularly distributive lattices). In a [[Distributive lattice|distributive lattice]], as in a commutative ring, all maximal ideals are prime; the converse implication holds in a [[Boolean algebra|Boolean algebra]], and indeed a distributive lattice in which all prime ideals are maximal is necessarily Boolean. As with rings, the set $  \mathop{\rm Specm}  L $
 +
of maximal ideals of a distributive lattice $  L $
 +
can be topologized as a subspace of the space $  \mathop{\rm Spec}  L $
 +
of all prime ideals, and it is a compact $  T _ {1} $-
 +
space; moreover, every compact $  T _ {1} $-
 +
space arises in this way. A distributive lattice $  L $
 +
is said to be normal if, given elements $  a , b \in L $
 +
with $  a \lor b = 1 $,  
 +
there exist $  c , d \in L $
 +
with $  a \lor d = c \lor b = 1 $
 +
and $  c \wedge d = 0 $.  
 +
Normal distributive lattices can be characterized as those for which every prime ideal is contained in a unique maximal ideal, or equivalently as those for which there is a continuous [[Retraction|retraction]] of $  \mathop{\rm Spec}  L $
 +
onto $  \mathop{\rm Specm}  L $;  
 +
they have the property that $  \mathop{\rm Specm}  L $
 +
is a Hausdorff space. For a topological space $  X $,  
 +
the lattice $  {\mathcal O} ( X) $
 +
of open subsets of $  X $
 +
is normal if and only if $  X $
 +
is a [[Normal space|normal space]]; if $  X $
 +
is a $  T _ {1} $-
 +
space, then $  \mathop{\rm Specm}  {\mathcal O} ( X) $
 +
yields a $  T _ {1} $[[
 +
Compactification|compactification]] of $  X $,  
 +
which coincides with the [[Stone–Čech compactification|Stone–Čech compactification]] if $  X $
 +
is normal (see [[Wallman compactification|Wallman compactification]]).
  
The construction of maximal ideals in arbitrary rings or lattices generally requires an appeal to Zorn's lemma (see [[Axiom of choice|Axiom of choice]] or [[Zorn lemma|Zorn lemma]]), and indeed the maximal ideal theorem for many classes of rings or lattices (i.e. the assertion that every non-trivial ring or lattice in the class has a maximal ideal) has been shown to be equivalent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296046.png" /> set theory to the axiom of choice. This applies to the class of all (commutative) unique factorization domains, and of all Heyting algebras (see [[Brouwer lattice|Brouwer lattice]]); however, for the classes of principal ideal domains, of Brouwer lattices, and of normal distributive lattices, the maximal ideal theorem is equivalent to the  "prime ideal theorem"  for the corresponding class, and is strictly weaker than the axiom of choice.
+
The construction of maximal ideals in arbitrary rings or lattices generally requires an appeal to Zorn's lemma (see [[Axiom of choice]] or [[Zorn lemma]]), and indeed the maximal ideal theorem for many classes of rings or lattices (i.e. the assertion that every non-trivial ring or lattice in the class has a maximal ideal) has been shown to be equivalent in $  \mathop{\rm ZF} $
 +
set theory to the axiom of choice. This applies to the class of all (commutative) unique factorization domains, and of all Heyting algebras (see [[Brouwer lattice]]); however, for the classes of principal ideal domains, of Brouwer lattices, and of normal distributive lattices, the maximal ideal theorem is equivalent to the  "[[Prime ideal theorem]]"  for the corresponding class, and is strictly weaker than the axiom of choice.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.T. Johnstone,  "Stone spaces" , Cambridge Univ. Press  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.T. Johnstone,  "Stone spaces" , Cambridge Univ. Press  (1983)</TD></TR></table>
  
In the theory of semi-groups (cf. [[Semi-group|Semi-group]]) maximal ideals play a lesser role than minimal ideals (cf. [[Minimal ideal|Minimal ideal]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296047.png" /> is a maximal two-sided ideal of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296048.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296050.png" /> is some indecomposable element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296051.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296052.png" />), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296053.png" /> is a prime ideal (that is, for any two ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296056.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296058.png" />). This implies that every maximal two-sided ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296059.png" /> is prime if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296060.png" />. In a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296061.png" /> with a maximal two-sided ideal a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296062.png" /> is maximal if (and, obviously, only if) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296063.png" /> contains the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296064.png" /> of all maximal two-sided ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296065.png" />. The Rees quotient semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296066.png" /> is an [[O-direct union|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296067.png" />-direct union]] of semi-groups each of which is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296068.png" />-simple or two-element nilpotent.
+
In the theory of semi-groups (cf. [[Semi-group|Semi-group]]) maximal ideals play a lesser role than minimal ideals (cf. [[Minimal ideal|Minimal ideal]]). If $  M $
 +
is a maximal two-sided ideal of a semi-group $  S $,  
 +
then either $  M = S \setminus  \{ a \} $,  
 +
where $  a $
 +
is some indecomposable element of $  S $(
 +
that is, $  a \in S  ^ {2} \setminus  S $),  
 +
or $  M $
 +
is a prime ideal (that is, for any two ideals $  A $
 +
and $  B $,  
 +
$  AB \subseteq M $
 +
implies $  A \subseteq M $
 +
or $  B \subseteq M $).  
 +
This implies that every maximal two-sided ideal in $  S $
 +
is prime if and only if $  S  ^ {2} = S $.  
 +
In a semi-group $  S $
 +
with a maximal two-sided ideal a prime ideal $  P \neq S $
 +
is maximal if (and, obviously, only if) $  P $
 +
contains the intersection $  I $
 +
of all maximal two-sided ideals of $  S $.  
 +
The Rees quotient semi-group $  S / I $
 +
is an [[O-direct union| $  O $-
 +
direct union]] of semi-groups each of which is either 0 $-
 +
simple or two-element nilpotent.
  
Sometimes a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296069.png" /> with proper left ideals may have a largest such ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296070.png" /> (that is, containing all other proper left ideals). This, for example, is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296071.png" /> has a right identity. In that case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296072.png" /> is not a singleton, then it is a sub-semi-group. In a periodic semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296073.png" /> the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296074.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062960/m06296075.png" /> is a (largest proper) two-sided ideal. Another example is given by subgroups with separating group part (see [[Invertible element|Invertible element]]) which is not a group.
+
Sometimes a semi-group $  S $
 +
with proper left ideals may have a largest such ideal $  L  ^ {*} $(
 +
that is, containing all other proper left ideals). This, for example, is the case when $  S $
 +
has a right identity. In that case, if $  S \setminus  L  ^ {*} $
 +
is not a singleton, then it is a sub-semi-group. In a periodic semi-group $  S $
 +
the existence of $  L  ^ {*} $
 +
implies that $  L  ^ {*} $
 +
is a (largest proper) two-sided ideal. Another example is given by subgroups with separating group part (see [[Invertible element|Invertible element]]) which is not a group.
  
 
====References====
 
====References====

Latest revision as of 08:00, 6 June 2020


A maximal element in the partially ordered set of proper ideals of a corresponding algebraic structure. Maximal ideals play an essential role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module $ M = R / I $ of $ R $ regarded as a left (respectively, right) $ R $- module relative to a left (respectively, right) maximal ideal $ I $ is irreducible (cf. Irreducible module); a homomorphism $ \phi $ of $ R $ into the field of endomorphisms of $ M $ is a representation of $ R $. The kernel of all such representations, that is, the set of elements of the ring which are mapped to zero by all representations, is called the Jacobson radical of $ R $; it coincides with the intersection of all maximal left (also, all right) ideals.

In the ring $ R = C [ a , b ] $ of continuous real-valued functions on a closed interval $ [ a , b ] $, the set of functions vanishing at some fixed point $ x _ {0} $ is a maximal ideal. Such ideals exhaust all maximal ideals of $ R $. This relation between the points of the interval and the maximal ideals has resulted in the construction of various theories for representing rings as rings of functions on a topological space.

The Zariski topology on the set of prime ideals (cf. Prime ideal) $ \mathop{\rm Spec} R $ of a ring $ R $ has weak separation properties (that is, there are non-closed points). A similar topology in the non-commutative case can be introduced on the set $ \mathop{\rm Spec} R $ of primitive ideals (cf. Primitive ideal), which are the annihilators of irreducible $ R $- modules. The set of maximal ideals, and in the non-commutative case, of maximal primitive ideals, forms a subspace $ \mathop{\rm Specm} R \subset \mathop{\rm Spec} R $ which satisfies the $ T _ {1} $- separation axiom.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)

Comments

Maximal ideals also play an important role in the structure and representation theory of lattices (particularly distributive lattices). In a distributive lattice, as in a commutative ring, all maximal ideals are prime; the converse implication holds in a Boolean algebra, and indeed a distributive lattice in which all prime ideals are maximal is necessarily Boolean. As with rings, the set $ \mathop{\rm Specm} L $ of maximal ideals of a distributive lattice $ L $ can be topologized as a subspace of the space $ \mathop{\rm Spec} L $ of all prime ideals, and it is a compact $ T _ {1} $- space; moreover, every compact $ T _ {1} $- space arises in this way. A distributive lattice $ L $ is said to be normal if, given elements $ a , b \in L $ with $ a \lor b = 1 $, there exist $ c , d \in L $ with $ a \lor d = c \lor b = 1 $ and $ c \wedge d = 0 $. Normal distributive lattices can be characterized as those for which every prime ideal is contained in a unique maximal ideal, or equivalently as those for which there is a continuous retraction of $ \mathop{\rm Spec} L $ onto $ \mathop{\rm Specm} L $; they have the property that $ \mathop{\rm Specm} L $ is a Hausdorff space. For a topological space $ X $, the lattice $ {\mathcal O} ( X) $ of open subsets of $ X $ is normal if and only if $ X $ is a normal space; if $ X $ is a $ T _ {1} $- space, then $ \mathop{\rm Specm} {\mathcal O} ( X) $ yields a $ T _ {1} $[[ Compactification|compactification]] of $ X $, which coincides with the Stone–Čech compactification if $ X $ is normal (see Wallman compactification).

The construction of maximal ideals in arbitrary rings or lattices generally requires an appeal to Zorn's lemma (see Axiom of choice or Zorn lemma), and indeed the maximal ideal theorem for many classes of rings or lattices (i.e. the assertion that every non-trivial ring or lattice in the class has a maximal ideal) has been shown to be equivalent in $ \mathop{\rm ZF} $ set theory to the axiom of choice. This applies to the class of all (commutative) unique factorization domains, and of all Heyting algebras (see Brouwer lattice); however, for the classes of principal ideal domains, of Brouwer lattices, and of normal distributive lattices, the maximal ideal theorem is equivalent to the "Prime ideal theorem" for the corresponding class, and is strictly weaker than the axiom of choice.

References

[a1] P.T. Johnstone, "Stone spaces" , Cambridge Univ. Press (1983)

In the theory of semi-groups (cf. Semi-group) maximal ideals play a lesser role than minimal ideals (cf. Minimal ideal). If $ M $ is a maximal two-sided ideal of a semi-group $ S $, then either $ M = S \setminus \{ a \} $, where $ a $ is some indecomposable element of $ S $( that is, $ a \in S ^ {2} \setminus S $), or $ M $ is a prime ideal (that is, for any two ideals $ A $ and $ B $, $ AB \subseteq M $ implies $ A \subseteq M $ or $ B \subseteq M $). This implies that every maximal two-sided ideal in $ S $ is prime if and only if $ S ^ {2} = S $. In a semi-group $ S $ with a maximal two-sided ideal a prime ideal $ P \neq S $ is maximal if (and, obviously, only if) $ P $ contains the intersection $ I $ of all maximal two-sided ideals of $ S $. The Rees quotient semi-group $ S / I $ is an $ O $- direct union of semi-groups each of which is either $ 0 $- simple or two-element nilpotent.

Sometimes a semi-group $ S $ with proper left ideals may have a largest such ideal $ L ^ {*} $( that is, containing all other proper left ideals). This, for example, is the case when $ S $ has a right identity. In that case, if $ S \setminus L ^ {*} $ is not a singleton, then it is a sub-semi-group. In a periodic semi-group $ S $ the existence of $ L ^ {*} $ implies that $ L ^ {*} $ is a (largest proper) two-sided ideal. Another example is given by subgroups with separating group part (see Invertible element) which is not a group.

References

[1a] S. Schwarz, "On maximal ideals in the theory of semigroups I" Czechoslovak. Math. J. , 3 (1953) pp. 139–153 (In Russian) (English abstract)
[1b] S. Schwarz, "On maximal ideals in the theory of semigroups II" Czechoslovak. Math. J. , 4 (1953) pp. 365–383 (In Russian) (English abstract)
[2] S. Schwarz, "Prime ideals and maximal ideals in semigroups" Czechoslovak. Math. J. , 19 (1969) pp. 72–79
[3] P.A. Grillet, "Intersections of maximal ideals in semigroups" Amer. Math. Monthly , 76 (1969) pp. 503–509

L.N. Shevrin

How to Cite This Entry:
Maximal ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_ideal&oldid=17927
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article