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Math Help - Ring homomorphism

  1. #1
    Member Maccaman's Avatar
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    Ring homomorphism

    Let  f : \mathbb{Z}/3\mathbb{Z} \rightarrow \ \mathbb{Z}/6\mathbb{Z} be defined by  f(x) = 4x Is f a well-defined ring homomorphism? If so, find its kernel and image.
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  2. #2
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    Quote Originally Posted by Maccaman View Post
    Let  f : \mathbb{Z}/3\mathbb{Z} \rightarrow \ \mathbb{Z}/6\mathbb{Z} be defined by  f(x) = 4x Is f a well-defined ring homomorphism? If so, find its kernel and image.
    yes it is and the kernel is (0). the easiest way to see this is to just check 3 elements of \mathbb{Z}/3\mathbb{Z} and see that f(x)=0 iff x=0. also you need to check that f(xy)=f(x)f(y), which is because

    16=4 in \mathbb{Z}/6\mathbb{Z}. the image is \{0,2,4 \}. (here by 0 i mean the coset 0 + 6 \mathbb{Z}, and by 2 i mean 2 + 6\mathbb{Z}, etc.)
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  3. #3
    Member Maccaman's Avatar
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    Thank-you for your help, but I'm still confused because I'm getting things mixed up. I'm having trouble seeing that  f is a well-defined ring homomorphism to start with. Could you show me why it is??
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