Difference between revisions of "Ring (mathematics)"

Revision as of 22:48, 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.

$(\mathbb {Z} ,+,{\dot {)}}$ - 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.
$(\mathbb {Z} /6\mathbb {Z} ,+,\cdot )$ : 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, $2\cdot 3=0$ , so, you can multiply two elements, neither of which is zero, and get zero as the result!

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.

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 '''4.''' Multiplication [[Distributivity|distribute]]s over addition.
+==Examples==
+*<math>(\mathbb{Z},+, \dot)</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.
+*<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!
 ==Remarks==