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Ring (mathematics) (view source)

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A '''ring''' in mathematics is a set ~~of numbers closed under addition and multiplication where addition is~~ [[~~commutative~~]]~~, both operations are~~ [[~~associative~~]], multiplication ~~[[Distributive property|distributes]] over addition~~, ~~there is an additive identity and additive inverses.~~

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A '''ring''' in [[mathematics]] is a set ''R'' equipped with two [[binary]] [[operations]], usually called addition and multiplication, satisfying that

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~~==Other defintions==~~

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~~Depending on the specific context, some elements of this definition may be somewhat different~~. ~~For example, some authors include in the defintion~~ that ~~there exists~~ a ~~multiplicative identity~~.

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'''1.''' ''R'' with addition is a [[commutative]] [[group]];

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'''2.''' ''R'' with multiplication is a [[semigroup]];

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'''3.''' multiplication [[distributes]] over addition.

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==Remarks==

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A '''ring with unit''' is a ring for which multiplication has a [[neutral element]].

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A '''division ring''' is a ring such that ''R'' with multiplication is a (not necessarily [[commutative]]) [[group]].

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