Difference between revisions of "Ring (mathematics)"

Revision as of 05:01, March 10, 2007

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 with multiplication is a semigroup;

3. multiplication distributes over addition.

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

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

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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.
+A '''ring''' in [[mathematics]] is a set ''R'' equipped with two [[binary]] [[operations]], usually called  addition and multiplication, satisfying that
-==Other defintions==
-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.
+'''1.''' ''R'' with addition is a [[commutative]] [[group]];
+'''2.''' ''R'' with multiplication is a [[semigroup]];
+'''3.''' multiplication [[distributes]] over addition.
+==Remarks==
+A '''ring with unit''' is a ring for which multiplication has a [[neutral element]].
+A '''division ring''' is a ring such that ''R'' with multiplication is a (not necessarily [[commutative]]) [[group]].