Difference between revisions of "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 of regarded as a left (respectively, right) -module relative to a left (respectively, right) maximal ideal is irreducible (cf. Irreducible module); a homomorphism of into the field of endomorphisms of is a representation of . 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 ; it coincides with the intersection of all maximal left (also, all right) ideals.
In the ring of continuous real-valued functions on a closed interval , the set of functions vanishing at some fixed point is a maximal ideal. Such ideals exhaust all maximal ideals of . 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) of a ring 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 of primitive ideals (cf. Primitive ideal), which are the annihilators of irreducible -modules. The set of maximal ideals, and in the non-commutative case, of maximal primitive ideals, forms a subspace which satisfies the -separation axiom.