R/A is a simple right R-module. Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals. Examples If F is a field, then the only maximal ideal is {0}. In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain. The ideal is a maximal ideal in ring . Generally, the maximal ideals of are of the form where is a prime number and is a polynomial in which is irreducible modulo . Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring whenever there exists an integer such that for any . The maximal ideals of the polynomial ring are principal ideals generated by for some . More generally, the maximal ideals of the polynomial ring over an algebraically closed field K are the ideals of the form . This result is known as the weak Nullstellensatz. Properties An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals. If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example, is a maximal ideal in , but is not a field. If L is a maximal left ideal, then R/L is a simple left R-module. Conversely in rings with unity, any simple left R-module arises this way. Incidentally this shows that a collection of representatives of simple left R-modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of R. Krull's theorem (1929): Every nonzero unital ring has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero finitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is a right ideal which is not R). Then R/I is a ring with unity (respectively, R/A is a finitely generated module), and so the above theorems can be applied