Unformatted text preview: concept out of the common characteristics of these examples, so we make the following deﬁnition: Deﬁnition 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 KŒX;Y Ł of polynomials in two variables over a ﬁeld K is a commutative ring....
View
Full
Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Transformations

Click to edit the document details