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Unformatted text preview: concept out of the common characteristics of these examples, so we make the following denition: Denition 1.11.1. A ring is a (nonempty) set R with two operations: addition, denoted here by C , and multiplication, denoted by juxtaposition, satisfying the following requirements: (a) Under addition, R is an abelian group. (b) Multiplication is associative. (c) Multiplication distributes over addition: a.b C c/ D ab C ac , and .b C c/a D ba C ca for all a;b;c 2 R . Multiplication need not be commutative in a ring, in general. If multiplication is commutative, the ring is called a commutative ring . Some additional examples of rings are as follows: Example 1.11.2. (a) The familiar number systems R , Q , C are rings. The set of natural numbers N is not a ring, because it is not a group under addition. (b) The set KX;Y of polynomials in two variables over a eld K is a commutative ring....
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 Fall '08
 EVERAGE
 Algebra, Transformations

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