Unformatted text preview: = 1, f ( 1 ) = 0, and f ( r ) = 0 otherwise; de f ne g : R → R by g ( ) = 0, g ( 1 ) = 1, and g ( r ) = 0 otherwise. Then neither f = 0 nor g = 0, but f g = 0. (iii) If R is a commutative ring, denote F ( R , R ) by F ( R ) : F ( R ) = { all functions R → R } . Show that F ( I 2 ) has exactly four elements, and that f + f = for every f ∈ F ( I 2 ) . Solution. There are exactly four functions from a 2point set to itself. If f ∈ F ( I 2 ) , then ( f + f )( r ) = f ( r ) + f ( r ) = for all r ∈ I 2 . 3.11 (i) Prove that the commutative ring C is a domain. Solution. If z w = 0 and z = a + ib ±= 0, then z ¯ z = a 2 + b 2 ±= 0, and µ ¯ z ¯ zz ¶ z = 1 . (ii) Prove that Z , Q , and R are domains. Solution. Every subring of a domain is a domain....
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 Fall '11
 KeithCornell
 Ring, Commutative ring

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