Editing Ring (mathematics) (section)

=== 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.