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Math Help - Abstract problems - rings

  1. #1
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    Abstract problems - rings

    Hi,

    1. Let E denote the set of even integers. Prove that with the usual addidion, and with multiplication defined by m*n=(1/2)mn, E is a ring. Is there a unity?

    2. Prove that aČ-bČ=(a+b)(a-b) for all a,b in a ring R iff R is commutative.

    I'm new to rings and am unsure how to get going with these problems. Any help would be great.

    Thanks.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by jackiemoon View Post
    Hi,

    1. Let E denote the set of even integers. Prove that with the usual addidion, and with multiplication defined by m*n=(1/2)mn, E is a ring. Is there a unity?
    well, check the following properties..
    1. Is E abelian under addition? (well, this is true for all integers/)
    1.5. Is the operation * well defined, i.e is a*b = (1/2)ab still an even integer?
    2. Is the operation * associative? a*(b*c)=(a*b)*c
    3. Is DPMA satisfied? a*(b+c) = a*b + a*c

    Quote Originally Posted by jackiemoon View Post
    2. Prove that aČ-bČ=(a+b)(a-b) for all a,b in a ring R iff R is commutative.

    I'm new to rings and am unsure how to get going with these problems. Any help would be great.

    Thanks.
    first, (a+b)(a-b) = (a+b)a - (a+b)b = a^2 + ba - ab -b^2 = a^2-b^2 \Longleftrightarrow ba-ab=0...
    Last edited by kalagota; March 1st 2009 at 04:11 AM.
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  3. #3
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    Hey Kalagota,

    Thanks for the reply. What does DPMA stand for?
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by jackiemoon View Post
    Hey Kalagota,

    Thanks for the reply. What does DPMA stand for?
    oh, Distributive Property of Multiplication over Addition..
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