Maximal ideal

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