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Unformatted text preview: 1.11. RINGS AND FIELDS 77 Example 1.11.5. (a) Z is a subring of Q . (b) The ring of 3by3 matrices with rational entries is a subring of the ring of 3by3 matrices with real entries. (c) The set of upper triangular 3by3 matrices with real entries is a subring of the ring of all 3by3 matrices with real entries. (d) The set of continuous functions from R to R is a subring of the ring of all functions from R to R . (e) The set of rationalvalued functions on a set X is a subring of the ring of realvalued functions on X . A second way in which two rings may be related to one another is by a homomorphism . A map f W R ! S between two rings is said to be a homomorphism if it respects the ring structures. More explicitly, f must take sums to sums and products to products. In notation, f.a C b/ D f.a/ C f.b/ and f.ab/ D f.a/f.b/ for all a;b 2 R . A bijective homomorphism is called an isomorphism ....
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 Fall '08
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 Algebra, Matrices

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