Editing Commutative ring (section)

==== Factor ring ====
The definition of ideals is such that "dividing" <math> I </math> "out" gives another ring, the ''factor ring'' <math> R / I </math>: it is the set of [[coset]]s of <math> I </math> together with the operations
<math display="block"> \left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I </math> and <math> \left(a+I\right) \left(b+I\right)=ab+I </math>.
For example, the ring <math> \mathbb{Z}/n\mathbb{Z} </math> (also denoted <math> \mathbb{Z}_n </math>), where <math> n </math> is an integer, is the ring of integers modulo <math> n </math>. It is the basis of [[modular arithmetic]].

An ideal is ''proper'' if it is strictly smaller than the whole ring. An ideal that is not strictly contained in any proper ideal is called [[maximal ideal|maximal]]. An ideal <math> m </math> is maximal [[if and only if]] <math> R / m </math> is a field. Except for the [[zero ring]], any ring (with identity) possesses at least one maximal ideal; this follows from [[Zorn's lemma]].