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A '''ring''' in [[mathematics]] is a set ''R'' equipped with two [[binary]] [[operation]]s, usually called addition and multiplication, that is a [[group]] under the operation of addition and a monoid (no inverses) under the operation of multiplication.

A '''ring''' in [[mathematics]] is a set ''R'' equipped with two [[binary]] [[operation]]s, usually called addition and multiplication, that is a [[group]] under the operation of addition and a monoid (no inverses) under the operation of multiplication.

==Examples==

==Examples==

2. '''Associativity''' - for any three elements <math>a,b,c\in R\ ,(a+b)+c = a+(b+c)</math>

2. '''Associativity''' - for any three elements <math>a,b,c\in R\ ,(a+b)+c = a+(b+c)</math>

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3. '''Inverses''' - for any element <math>a \in R</math> there exists an element (usually labeled as <math>-a</math> such that <math>a+(-a) = (-a)+a = 0</math>

3. '''Inverses''' - for any element <math>a \in R</math> there exists an element (usually labeled as <math>-a</math> such that <math>a+(-a) = (-a)+a = 0</math>)

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4. ''''Commutativity''' - for any two elements <math>a,b \in R\ ,a+b = b+a</math>

4. ''''Commutativity''' - for any two elements <math>a,b \in R\ ,a+b = b+a</math>