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Unformatted text preview: 76 1. ALGEBRAIC THEMES (c) The set of real–valued functions on a set X , with pointwise addi tion and multiplication of functions, .f C g/.x/ D f.x/ C g.x/ , and .fg/.x/ D f.x/g.x/ , for functions f and g and x 2 X , is a commutative ring. (d) The set of nby n matrices with integer entries is a noncommu tative ring. There are many, many variations on these examples: matrices with complex entries or with with polynomial entries; functions with complex values, or integer values; continuous or differentiable functions; polyno mials with any number of variables. Many rings have an identity element for multiplication , usually de noted by 1 . (The identity element satisfies 1a D a1 D a for all elements a of the ring. Some rings do not have an identity element!) Example 1.11.3. (a) The identity matrix (diagonal matrix with diagonal entries equal to 1) is the multiplicative identity in the nby n matrices....
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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, Addition, Multiplication

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