= Ring
{disambiguate=mathematics}
{wiki}
= Ring
{synonym}
A <Ring (mathematics)> can be seen as a generalization of a <field (mathematics)> where:
* multiplication is not necessarily <commutative>. If this is satisfied, we can call it a <commutative ring>.
* multiplication may not have <inverse elements>. If this is satisfied, we can call it a <division ring>.
Addition however has to be <commutative> and have inverses, i.e. it is an <Abelian group>.
The simplest example of a ring which is not a full fledged <field (mathematics)> and with <commutative> multiplication are the <integers>. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the <set (mathematics)>. More specifically, the <integers> are a <commutative ring>.
A <polynomial ring> is another example with the same properties as the <integers>.
The simplest <commutative ring>[non-commutative], <division ring>[non-division] is is the set of all 2x2 <matrices> of <real numbers>:
* we know that 2x2 matrix multiplication is non-commutative in general
* some 2x2 matrices have a multiplicative inverse, but others don't
Note that <GL(n)> is not a ring because you can by addition reach the zero matrix.
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