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Unformatted text preview: 266 6. RINGS Example 6.1.4. Let p.x;y;z/ D 7xyz C 3x 2 yz 2 C 2yz 3 and q.x;y;z/ D 2 C 3xz C 2xyz . Then p.x;y;z/ C q.x;y;z/ D 2 C 3xz C 9xyz C 3x 2 yz 2 C 2yz 3 ; and p.x;y;z/q.x;y;z/ D 14x y z C 27x 2 y z 2 C 14x 2 y 2 z 2 C 4y z 3 C 9x 3 y z 3 C 6x 3 y 2 z 3 C 6x y z 4 C 4x y 2 z 4 : 3. Rings of functions. Let X be any set and let R be a field. Then the set of functions defined on X with values in R is a ring, with the operations defined pointwise: .f C g/.x/ D f.x/ C g.x/ , and .fg/.x/ D f.x/g.x/ . If X is a metric space (or a topological space) and R is equal to one of the fields R or C , then the set of continuous R valued functions on X , with pointwise operations, is a ring. The essential point here is that the sum and product of continuous functions are continuous. (If you are not familiar with metric or topological spaces, just think of X as a subset of R .) If X is an open subset of C , then the set of holomorphic C valued functions on X is a ring. (If you are not familiar with holomorphic func-is a ring....
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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