Editing Ring (mathematics) (section)

== Notes ==
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{{cnote|a|Some authors only require that a ring be a [[semigroup]] under multiplication; that is, do not require that there be a multiplicative identity (1). See the section {{slink|#Variations on terminology}} for more details.}}
{{cnote|b|Elements which do have multiplicative inverses are called [[unit (ring theory)|units]], see {{harvnb|Lang|2002|loc=§II.1|p=84}}.}}
{{cnote|c|The closure axiom is already implied by the condition that +/⋅ be a binary operation. Some authors therefore omit this axiom. {{harvnb|Lang|2002}}}}
{{cnote|d|The transition from the integers to the rationals by adding fractions is generalized by the [[quotient field]].}}
{{cnote|e|Many authors include [[commutative ring|commutativity of rings]] in the set of ''ring axioms'' ([[Ring (mathematics)#Formal definition|see above]]) and therefore refer to "commutative rings" as just "rings".}}
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