Editing Commutative ring (section)

== Divisibility ==
In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of [[divisibility (ring theory)|divisibility for rings]] is richer. An element <math> a </math> of ring <math> R </math> is called a [[unit (algebra)|unit]] if it possesses a multiplicative inverse. Another particular type of element is the [[zero divisor]]s, i.e. an element <math> a </math> such that there exists a non-zero element <math> b </math> of the ring such that <math> ab = 0 </math>. If <math> R </math> possesses no non-zero zero divisors, it is called an [[integral domain]] (or domain). An element <math> a </math> satisfying <math> a^n = 0  </math> for some positive integer <math> n </math> is called [[nilpotent element|nilpotent]].

=== Localizations ===
{{Main|Localization of a ring}}
The ''localization'' of a ring is a process in which some elements are rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if <math> S </math> is a [[multiplicatively closed subset]] of <math> R </math> (i.e. whenever <math> s,t \in S </math> then so is <math> st </math>) then the ''localization'' of <math> R </math> at <math> S </math>, or ''ring of fractions'' with denominators in <math> S </math>, usually denoted <math> S^{-1}R </math> consists of symbols
{{block indent|1= <math>\frac{r}{s}</math> with <math> r \in R, s \in S </math> }}
subject to certain rules that mimic the cancellation familiar from rational numbers. Indeed, in this language <math> \mathbb{Q} </math> is the localization of <math> \mathbb{Z} </math> at all nonzero integers. This construction works for any integral domain <math> R </math> instead of <math> \mathbb{Z} </math>. The localization <math> \left(R\setminus \left\{0\right\}\right)^{-1}R </math> is a field, called the [[quotient field]] of <math> R </math>.