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Math Help - Algebra over a field - zero element

  1. #1
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    Algebra over a field - zero element

    A is an algebra over a field F.

    Let 'a' in A have the following property:
    a.b = 0 for all 'b' in A.

    Does this imply a=0? If yes why?
    Thanks,
    Aman

    PS: I was trying the above for a ring. I am kind of convinced that in rings 'a' need not be equal to 0. Haven't been able to find a concrete example though.
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  2. #2
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    Quote Originally Posted by aman_cc View Post
    A is an algebra over a field F.

    Let 'a' in A have the following property:
    a.b = 0 for all 'b' in A.

    Does this imply a=0? If yes why?
    Thanks,
    Aman

    PS: I was trying the above for a ring. I am kind of convinced that in rings 'a' need not be equal to 0. Haven't been able to find a concrete example though.
    It is true in both fields and rings with identity that "if ab= 0 for all b in A then a= 0". Just take a times the identity, i. ai= a= 0. To find a counter-example, you would need to look at rings without identity.

    In a field, it is true that if ab= 0 for some non-zero b (not necessarily "all" b) then a= 0. If b is non-zero, it has an inverse, so abb^{-1}= a= 0b^{-1}= 0.
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  3. #3
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    Thanks. But what about a more general case? Fields will always have 1. So they are not a problem.

    What about 1. Algebra 2. Ring without and multiplicative identity,1.
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  4. #4
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    Quote Originally Posted by aman_cc View Post
    Thanks. But what about a more general case? Fields will always have 1. So they are not a problem.

    What about 1. Algebra 2. Ring without and multiplicative identity,1.
    Guess for rings following example will do:
    R = {0,2}

    + is defined as (a+b) mod 4
    . is defines as (a.b) mod 4

    Under these 2.a = 0 for all a. Yet 2 =/= 0.
    Is this correct?

    Can you help for Algebra, A in the original question?
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  5. #5
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    Quote Originally Posted by aman_cc View Post
    Guess for rings following example will do:
    R = {0,2}

    + is defined as (a+b) mod 4
    . is defines as (a.b) mod 4

    Under these 2.a = 0 for all a. Yet 2 =/= 0.
    Is this correct?

    Can you help for Algebra, A in the original question?
    your example is correct. an example for algebras: consider any vector space A \neq (0) over a field F and define multiplication in A by xy=0, for all x,y \in A.
    Last edited by NonCommAlg; September 9th 2009 at 04:56 AM.
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  6. #6
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    "for an algebra A over a field F, note that A contains (a copy) of F (in its center) and thus A always has an identity element"

    Haven't really understood the above point?
    Thanks
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  7. #7
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    Quote Originally Posted by aman_cc View Post
    "for an algebra A over a field F, note that A contains (a copy) of F (in its center) and thus A always has an identity element"

    Haven't really understood the above point?
    Thanks
    sorry, i made a mistake! what i wrote is correct if A has an identity element. then A contains a copy of F in its center and 1_A=1_F. now see my previous post that i just edited.
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