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Multiplication modulo algebra

  1. Feb 20, 2014 #1
    Question: Let n be a positive integer and consider x,y,z to be elements of Zn. Prove that if x . y = 1 and x . z = 1, then y = z. (Since working in Zn the sign '.' means "multiplication modulo n".)

    Conclude that if x has an inverse element in Zn, then the inverse element is unique.

    Attempt: Well if x . y = 1 then this means that xy = n + 1 since were working in Zn. This means the same can be said about y . z = 1 that yz = n + 1. This suggests that z = x by substitution. I'm not sure of this is correct or if I'm going down right path here.

    Other thing I thought about was multiplying x . y = 1 by z on both sides to prove y=z but I'm a bit stuck.

    I'm also a bit confused about the last bit of the question where I have to conclude if x has inverse element (not entirely sure what this is) in Zn then the inverse element is unique.

    Can anyone help please?

    Sent from my iPhone using Physics Forums
  2. jcsd
  3. Feb 20, 2014 #2


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    This is NOT true. for example, 5*5= 1 (mod 12) because 5*5= 2(12+ 1). That is, 2n+ 1, NOT n+ 1.

    Why "a bit stuck"? Where did you get stuck? Since multiplication is commutative, yes, if xy= 1 then zxy= (xz)y= z so y= z.

    An "inverse element" of x is another element, y, such that xy= yx= 1. For any "other inverse element", z, xz= 1.

    I'm a bit puzzled why, on recognizing that you did not know the definitions well ("not entirely sure what this is"), you did not immediately look up and review the definitions.
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