# 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.