Difference between revisions of "Ring (mathematics)"
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| − | A '''ring''' in [[mathematics]] is a set ''R'' equipped with two [[binary]] [[operation]]s, usually called | + | 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'' | + | '''2.''' ''R'' is closed under multiplication; |
| − | '''3.''' | + | '''3.''' Multiplication is [[associative]]; |
| + | |||
| + | '''4.''' Multiplication [[Distributivity|distribute]]s over addition. | ||
==Remarks== | ==Remarks== | ||
| − | A '''ring with | + | 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]] | ||
Revision as of 22:14, November 14, 2008
A ring in mathematics is a set R equipped with two binary operations, usually called addition and multiplication, satisfying that
1. R with addition is a commutative group;
2. R is closed under multiplication;
3. Multiplication is associative;
4. Multiplication distributes over addition.
Remarks
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.