Editing Maximal ideal (section)

==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 ideal]]s generated by a prime number.
* More generally, all nonzero [[prime ideal]]s are maximal in a [[principal ideal domain]].
* The ideal <math> (2, x) </math> is a maximal ideal in ring <math> \mathbb{Z}[x] </math>. Generally, the maximal ideals of <math> \mathbb{Z}[x] </math> are of the form <math> (p, f(x)) </math> where <math> p </math> is a prime number and <math> f(x) </math> is a polynomial in <math> \mathbb{Z}[x] </math> which is irreducible modulo <math> p </math>.
* 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 <math> R </math> whenever there exists an integer <math> n > 1 </math> such that <math> x^n = x </math> for any <math> x \in R </math>.
* The maximal ideals of the [[polynomial ring]] <math>\mathbb{C}[x]</math> are principal ideals generated by <math>x-c</math> for some <math>c\in \mathbb{C}</math>.
* More generally, the maximal ideals of the polynomial ring {{nowrap|''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} over an [[algebraically closed field]] ''K'' are the ideals of the form {{nowrap|(''x''<sub>1</sub>&nbsp;&minus;&nbsp;''a''<sub>1</sub>, ..., ''x''<sub>''n''</sub>&nbsp;&minus;&nbsp;''a''<sub>''n''</sub>)}}. This result is known as the weak [[Nullstellensatz]].