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Math Help - Abstract algebra - associative commutative binary operations?

  1. #1
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    Abstract algebra - associative commutative binary operations?

    Show that *,defined on Q (where Q is a set of rational numbers) by

    a * b = a + b + 3ab

    is a commutative binary operation.Is is associative?
    Ive answered this part but i dont know how to answer part b

    Determine the identity element admitted by * and show that it is unique.Show that with respect to this identity the inverse a(^ -1) of a is given by

    a(^ -1) = ((-a)/(1+3a))

    Give an element a C Q which has no inverse with respect to *.
    (where C means contained, Q is a set of rational numbers)
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  2. #2
    Senior Member Shanks's Avatar
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    If we denote the identity by e, then for any a in Q:
    we have a*e=a+e+3ae=a
    thus e=0.
    a*a^{-1}=e=0 gives the formula of the inverse of a.
    a has no inverse iff 1+3a=0.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Shanks View Post
    a*a^{-1}=e=0 gives the formula of the inverse of a.
    a has no inverse iff 1+3a=0.
    EDIT: Never mind. I miss-read your post.
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  4. #4
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    Dear asingh88,
    Please see the attachments given below. I have given the complete answer. If you have any questions please do'nt hesitate to ask me.
    Attached Thumbnails Attached Thumbnails Abstract algebra - associative commutative binary operations?-sol1.jpg   Abstract algebra - associative commutative binary operations?-sol2.jpg  
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