Difference between revisions of "Ring (mathematics)"

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'''1.''' ''R'' with addition is a [[commutative]] [[group]];  
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'''1.''' ''R'' with addition is a [[commutative]] [[Group (mathematics)|group]];  
  
 
'''2.''' ''R'' is closed under multiplication;
 
'''2.''' ''R'' is closed under multiplication;
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==Examples==
 
==Examples==
*<math>(\mathbb{Z},+, \cdot)</math> - the set of the integers - together with the usual [[addition]] and [[multiplication]] is a ring.
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*<math>(\mathbb{Z},+, \cdot)</math> - the set of the [[integers]] - together with the usual [[addition]] and [[multiplication]] is a ring.
 
*the [[subring]] of the even numbers is a ring, too: this shows that there is not necessarily a neutral element of the multiplication in a ring.
 
*the [[subring]] of the even numbers is a ring, too: this shows that there is not necessarily a neutral element of the multiplication in a ring.
 
*<math>(\mathbb{Z} / 6\mathbb{Z},+, \cdot)</math> : this is the ring of six elements {0,1,2,3,4,5} and the usual addition and multiplication [[modulo]] six. So, here 1+3= 4, but 4+5 = 3. Interestingly, <math>2 \cdot 3 = 0 </math>, so, you can multiply two elements, neither of which is zero, and get zero as the result!
 
*<math>(\mathbb{Z} / 6\mathbb{Z},+, \cdot)</math> : this is the ring of six elements {0,1,2,3,4,5} and the usual addition and multiplication [[modulo]] six. So, here 1+3= 4, but 4+5 = 3. Interestingly, <math>2 \cdot 3 = 0 </math>, so, you can multiply two elements, neither of which is zero, and get zero as the result!
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A '''commutative ring''' is a ring in which multiplication is commutative.
 
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]].
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A '''division ring''' is a ring such that ''R'' with multiplication is a (not necessarily [[commutative]]) [[Group (mathematics)|group]].
  
 
[[Category:Algebra]]
 
[[Category:Algebra]]

Revision as of 21:01, November 17, 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.

Examples

  • - the set of the integers - together with the usual addition and multiplication is a ring.
  • the subring of the even numbers is a ring, too: this shows that there is not necessarily a neutral element of the multiplication in a ring.
  •  : this is the ring of six elements {0,1,2,3,4,5} and the usual addition and multiplication modulo six. So, here 1+3= 4, but 4+5 = 3. Interestingly, , so, you can multiply two elements, neither of which is zero, and get zero as the result!

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.