Changes

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

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

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

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

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

'''2.''' ''R'' is closed under multiplication;

'''3.''' multiplication [[distribute]]s over addition.

'''3.''' Multiplication is [[associative]];

 

'''4.''' Multiplication [[Distributivity|distribute]]s over addition.

==Remarks==

==Remarks==

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

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

 

A '''commutative ring''' is a ring in which multiplication is commutative.

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

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

[[Category:Algebra]]

[[Category:Algebra]]