Editing Maximal ideal (section)

{{Short description|Ideal of a ring contained in no other ideal except the ring itself}}
In [[mathematics]], more specifically in [[ring theory]], a '''maximal ideal''' is an [[ideal (ring theory)|ideal]] that is [[maximal element|maximal]] (with respect to [[set inclusion]]) amongst all ''proper'' ideals.<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref> In other words, ''I'' is a maximal ideal of a [[ring (mathematics)|ring]] ''R'' if there are no other ideals contained between ''I'' and ''R''.

Maximal ideals are important because the [[Quotient ring|quotients of rings]] by maximal ideals are [[simple ring]]s, and in the special case of [[Ring_(mathematics)#Notes_on_the_definition|unital]] [[commutative ring]]s they are also [[field (mathematics)|field]]s.  The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the [[Zariski topology]], is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. 

In noncommutative ring theory, a '''maximal right ideal''' is defined analogously as being a maximal element in the [[poset]] of proper right ideals, and similarly, a '''maximal left ideal''' is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a [[simple module]] over ''R''.  If ''R'' has a unique maximal right ideal, then ''R'' is known as a [[local ring]], and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the [[Jacobson radical]] J(''R'').

It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example, in the ring of 2 by 2 [[square matrices]] over a field, the [[zero ideal]] is a maximal two-sided ideal, but there are many maximal right ideals.