Editing Ring (mathematics) (section)

== Definition ==
A '''ring''' is a [[Set (mathematics)|set]] {{mvar|R}} equipped with two binary operations{{efn|This means that each operation is defined and produces a unique result in {{mvar|R}} for each ordered pair of elements of {{mvar|R}}.}} + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the '''ring axioms''':{{sfnp|Bourbaki|1989|p=96|loc=Ch 1, §8.1|ps=}}{{sfnp|Mac Lane|Birkhoff|1967|p=85|ps=}}{{sfnp|Lang|2002|p=83|ps=}}
# {{mvar|R}} is an [[abelian group]] under addition, meaning that:
#* {{math|1=(''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'')}} for all {{math|''a'', ''b'', ''c''}} in {{mvar|R}} (that is, {{math|+}} is [[associativity|associative]]).
#* {{math|1=''a'' + ''b'' = ''b'' + ''a''}} for all {{math|''a'', ''b''}} in {{mvar|R}} (that is, {{math|+}} is [[commutativity|commutative]]).
#* There is an element {{math|0}} in {{mvar|R}} such that {{math|1=''a'' + 0 = ''a''}} for all {{mvar|a}} in {{mvar|R}} (that is, {{math|0}} is the [[additive identity]]).
#* For each {{mvar|a}} in {{mvar|R}} there exists {{math|−''a''}} in {{mvar|R}} such that {{math|1=''a'' + (−''a'') = 0}} (that is, {{math|−''a''}} is the [[additive inverse]] of {{mvar|a}}).
# {{mvar|R}} is a [[monoid]] under multiplication, meaning that:
#* {{math|1=(''a'' · ''b'') · ''c'' = ''a'' · (''b'' · ''c'')}} for all {{math|''a'', ''b'', ''c''}} in {{mvar|R}} (that is, {{math|⋅}} is associative).
#* There is an element {{math|1}} in {{mvar|R}} such that {{math|1=''a'' · 1 = ''a''}} and {{math|1=1 · ''a'' = ''a''}} for all {{mvar|a}} in {{mvar|R}} (that is, {{math|1}} is the [[multiplicative identity]]).{{efn|The existence of 1 is not assumed by some authors; here, the term ''[[rng (algebra)|rng]]'' is used if existence of a multiplicative identity is not assumed.<!-- This is the most common convention, and is adopted throughout wikipedia, please do not change --> See [[Ring (mathematics)#Variations on terminology|next subsection]].}}
# Multiplication is [[distributive law|distributive]] with respect to addition, meaning that:
#* {{math|1=''a'' · (''b'' + ''c'') = (''a'' · ''b'') + (''a'' · ''c'')}} for all {{math|''a'', ''b'', ''c''}} in {{mvar|R}} (left distributivity).
#* {{math|1=(''b'' + ''c'') · ''a'' = (''b'' · ''a'') + (''c'' · ''a'')}} for all {{math|''a'', ''b'', ''c''}} in {{mvar|R}} (right distributivity).

In notation, the multiplication symbol {{math|·}} is often omitted, in which case {{math|''a'' · ''b''}} is written as {{math|''ab''}}.

=== Variations on terminology ===
In the terminology of this article, a ring is defined to have a multiplicative identity,<!--- This is also the convention in [[Wikipedia:Manual of Style/Mathematics]]. ---> while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a <!--- What follows is not a typo: it is intentionally "rng". --->"[[rng (algebra)|{{not a typo|rng}}]]" (IPA: {{IPAc-en|r|ʊ|ŋ}}) with a missing "i".  For example, the set of [[even integer]]s with the usual + and ⋅ is a rng, but not a ring.  As explained in ''{{section link||History}}'' below, many authors apply the term "ring" without requiring a multiplicative identity.

Although ring addition is [[commutative law|commutative]], ring multiplication is not required to be commutative: {{mvar|ab}} need not necessarily equal {{math|''ba''}}. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called ''[[commutative ring]]s''. Books on commutative algebra or algebraic geometry often adopt the convention that ''ring'' means ''commutative ring'', to simplify terminology.

In a ring, multiplicative inverses are not required to exist. A non[[zero ring|zero]] commutative ring in which every nonzero element has a [[multiplicative inverse]] is called a [[field (mathematics)|field]].

The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.{{sfnp|Isaacs|1994|p=160|ps=}} The proof makes use of the "{{math|1}}", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: {{math|1=''ab'' + ''cd'' = ''cd'' + ''ab''}}.)

There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative.<ref>{{cite web |url=https://www.encyclopediaofmath.org/index.php/Non-associative_rings_and_algebras |title=Non-associative rings and algebras |website=Encyclopedia of Mathematics}}</ref> For these authors, every [[algebra over a field|algebra]] is a "ring".