Difference between revisions of "Ring"
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| − | A '''ring''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements with two | + | A '''ring''' is a structure of [[abstract algebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements with two [[operation]]s, usually called multiplication and addition and denoted <math>\cdot</math> and <math>+</math>, which have the following properties: |
| − | Under the operation +, the ring is an [[abelian group]] and so obeys all the group axioms (existence of an [[identity]], existence of [[inverse]] | + | Under the operation +, the ring is an [[abelian group]] and so obeys all the group axioms (existence of an [[identity]], existence of [[inverse with respect to an operation | inverses]], [[associative | associativity]]) as well as [[commutative | commutivity]]. |
There exists an element, usually denoted 1, such that <math>1 \cdot a = a \cdot 1 = a</math> for all <math>a\in R</math>. (Multiplicative identity.) | There exists an element, usually denoted 1, such that <math>1 \cdot a = a \cdot 1 = a</math> for all <math>a\in R</math>. (Multiplicative identity.) | ||
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For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b\cdot c) = (a\cdot b)\cdot c</math>. (Associativity.) | For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b\cdot c) = (a\cdot b)\cdot c</math>. (Associativity.) | ||
| − | For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b + c) = (a\cdot b) + (a\cdot c)</math> and <math>(a + b)\cdot c = (a\cdot c) + (b\cdot c)</math>. (Distributivity of multiplication over addition.) | + | For every three elements <math>a, b, c\in R</math> we have <math>a\cdot(b + c) = (a\cdot b) + (a\cdot c)</math> and <math>(a + b)\cdot c = (a\cdot c) + (b\cdot c)</math>. ([[Distributive property | Distributivity]] of multiplication over addition.) |
| − | Note especially that multiplicative inverses need not exist | + | Note especially that multiplicative inverses need not exist and that multiplication need not be commutative. |
Common examples of rings include the [[integer]]s or the integers taken [[modular arithmetic|modulo]] <math>n</math>, with addition and multiplication as usual. In addition, every field is a ring. | Common examples of rings include the [[integer]]s or the integers taken [[modular arithmetic|modulo]] <math>n</math>, with addition and multiplication as usual. In addition, every field is a ring. | ||
Revision as of 08:24, 18 July 2006
This article is a stub. Help us out by expanding it.
A ring is a structure of abstract algebra, similar to a group or a field. A ring 
is a set of elements with two operations, usually called multiplication and addition and denoted 
and 
, which have the following properties:
Under the operation +, the ring is an abelian group and so obeys all the group axioms (existence of an identity, existence of inverses, associativity) as well as commutivity.
There exists an element, usually denoted 1, such that 
for all 
. (Multiplicative identity.)
For every three elements 
we have 
. (Associativity.)
For every three elements we have 
and 
. ( Distributivity of multiplication over addition.)
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
Common examples of rings include the integers or the integers taken modulo 
, with addition and multiplication as usual. In addition, every field is a ring.
