Editing Maximal ideal (section)

==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, <math>4\mathbb{Z}</math> is a maximal ideal in <math>2\mathbb{Z} </math>, but <math>2\mathbb{Z}/4\mathbb{Z}</math> 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 to the quotient to conclude that there is a maximal ideal (respectively, maximal right ideal) of ''R'' containing ''I'' (respectively, ''A'').
* Krull's theorem can fail for rings without unity. A [[radical ring]], i.e. a ring in which the [[Jacobson radical]] is the entire ring, has no simple modules and hence has no maximal right or left ideals. See [[regular ideal]]s for possible ways to circumvent this problem.
* In a commutative ring with unity, every maximal ideal is a [[prime ideal]]. The converse is not always true: for example, in any nonfield [[integral domain]] the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as [[Commutative ring#Dimension|zero-dimensional rings]], where the dimension used is the [[Krull dimension]].
* If ''k'' is a field, the preimage of a maximal ideal of a [[Finitely generated algebra|finitely generated ''k''-algebra]] under a ''k''-algebra homomorphism is a maximal ideal.  However, the preimage of a maximal ideal of a unital commutative ring under a ring homomorphism is not necessarily maximal.  For example, let <math>f:\mathbb{Z}\to\mathbb{Q}</math> be the inclusion map and <math>\mathfrak{n}=(0)</math> in <math>\mathbb{Q}</math>.  Then <math>\mathfrak{n}</math> is maximal in <math>\mathbb{Q}</math> but <math>f^{-1}(\mathfrak{n})=(0)</math> is not maximal in <math>\mathbb{Z}</math>.
* A maximal ideal of a noncommutative ring might not be prime in the commutative sense. For example, let <math>M_{n\times n}(\mathbb{Z})</math> be the ring of all <math>n\times n</math> matrices over <math>\mathbb{Z}</math>. This ring has a maximal ideal <math>M_{n\times n}(p\mathbb{Z})</math> for any prime <math>p</math>, but this is not a prime ideal since (in the case <math>n=2</math>)<math>A=\text{diag}(1,p)</math> and <math>B=\text{diag}(p,1)</math> are not in <math>M_{n\times n}(p\mathbb{Z})</math>, but <math>AB=pI_2\in M_{n\times n}(p\mathbb{Z})</math>. However, maximal ideals of noncommutative rings ''are'' prime in the [[#Generalization|generalized sense]] below.