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Unformatted text preview: Math 3124 Tuesday, November 15 November 15, Ungraded Homework Problem 27.14 on page 135. What can be said about the characteristic of a ring R in which x = x for each x ∈ R ? Since x = x for all x ∈ R , by adding x to both sides we see that 2 x = 0 for all x ∈ R . Therefore the characteristic of R is at most 2; in other words it is 1 or 2. The characteristic being 1 means that R = { } . We conclude that either the characteristic of R is 2 or R is the zero ring. Section 38 on page 180. Which of the mappings in the following problems are ring homo morphisms? Determine the kernel of each mapping that is a homomorphism. 1. θ : Z → Z by θ ( a ) = 3 a This is not a homomorphism because θ does not respect multiplication. For example θ ( 1 * 1 ) = θ ( 1 ) = 3, yet θ ( 1 ) * θ ( 1 ) = 3 * 3 = 9. 2. θ : Z → Z by θ ( a ) = a 2 This is not a homomorphism because θ does not respect addition. For example we have θ ( 1 + 1 ) = ( 1 + 1 ) 2 = 4, yet θ ( 1 )+ θ ( 1 ) = 1 + 1 = 2.2....
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 Fall '08
 PARRY
 Math, Algebra

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