Editing Ring (mathematics) (section)

== Illustration ==
[[File:Number-line.svg|alt=|thumb|410x410px|The [[integer]]s, along with the two operations of [[addition]] and [[multiplication]], form the prototypical example of a ring.]]

The most familiar example of a ring is the set of all integers {{tmath|\Z,}} consisting of the [[number]]s
: <math>\dots,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots</math>

The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.

=== Some properties ===
Some basic properties of a ring follow immediately from the axioms:
* The additive identity is unique.
* The additive inverse of each element is unique.
* The multiplicative identity is unique.
* For any element {{mvar|x}} in a ring {{mvar|R}}, one has {{math|1=''x''0 = 0 = 0''x''}} (zero is an [[absorbing element]] with respect to multiplication) and {{math|1=(–1)''x'' = –''x''}}.
* If {{math|1=0 = 1}} in a ring {{mvar|R}} (or more generally, {{math|0}} is a unit element), then {{mvar|R}} has only one element, and is called the [[zero ring]].
* If a ring {{mvar|R}} contains the zero ring as a subring, then {{mvar|R}} itself is the zero ring.{{sfnp|Isaacs|1994|p=161|ps=}}
* The [[binomial formula]] holds for any {{mvar|x}} and {{mvar|y}} satisfying {{math|1=''xy'' = ''yx''}}.

=== Example: Integers modulo 4 ===
{{see also| Modular arithmetic}}

Equip the set <math>\Z /4\Z = \left\{\overline{0}, \overline{1}, \overline{2}, \overline{3}\right\}</math> with the following operations:
* The sum <math>\overline{x} + \overline{y}</math> in {{tmath|\Z/4\Z}} is the remainder when the integer {{math|''x'' + ''y''}} is divided by {{math|4}} (as {{math|''x'' + ''y''}} is always smaller than {{math|8}}, this remainder is either {{math|''x'' + ''y''}} or {{math|1=''x'' + ''y'' − 4}}). For example, <math>\overline{2} + \overline{3} = \overline{1}</math> and <math>\overline{3} + \overline{3} = \overline{2}.</math>
* The product <math>\overline{x} \cdot \overline{y}</math> in {{tmath|\Z/4\Z}} is the remainder when the integer {{mvar|xy}} is divided by {{math|4}}. For example, <math>\overline{2} \cdot \overline{3} = \overline{2}</math> and <math>\overline{3} \cdot \overline{3} = \overline{1}.</math>

Then {{tmath|\Z/4\Z}} is a ring: each axiom follows from the corresponding axiom for {{tmath|\Z.}} If {{mvar|x}} is an integer, the remainder of {{mvar|x}} when divided by {{math|4}} may be considered as an element of {{tmath|\Z/4\Z,}} and this element is often denoted by "{{math|''x'' mod 4}}" or <math>\overline x,</math> which is consistent with the notation for {{math|0, 1, 2, 3}}. The additive inverse of any <math>\overline x</math> in {{tmath|\Z/4\Z}} is <math>-\overline x=\overline{-x}.</math> For example, <math>-\overline{3} = \overline{-3} = \overline{1}.</math>

{{tmath|\Z/4\Z}} has a subring {{tmath|\Z/2\Z}}, and if <math>p</math> is prime, then {{tmath|\Z/p\Z}} has no subrings.

=== Example: 2-by-2 matrices ===
The set of 2-by-2 [[square matrices]] with entries in a [[field (mathematics)|field]] {{mvar|F}} is{{sfnp|Lam|2001|loc=Theorem 3.1|ps=}}{{sfnp|Lang|2005|loc=Ch V, §3}}{{sfnp|Serre|2006|p=3|ps=}}{{sfnp|Serre|1979|p=158|ps=}}
: <math>\operatorname{M}_2(F) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\  a, b, c, d \in F \right\}.</math>

With the operations of matrix addition and [[matrix multiplication]], <math>\operatorname{M}_2(F)</math> satisfies the above ring axioms. The element <math>\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)</math> is the multiplicative identity of the ring. If <math>A = \left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)</math> and <math>B = \left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right),</math> then <math>AB = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix} \right)</math> while <math>BA = \left( \begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix} \right);</math> this example shows that the ring is noncommutative.

More generally, for any ring {{mvar|R}}, commutative or not, and any nonnegative integer {{mvar|n}}, the square {{math|''n'' × ''n''}} matrices with entries in {{mvar|R}} form a ring; see ''[[Matrix ring]]''.