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Ring theory- zero divisors and integral domains

  1. Feb 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the ring Z/mZ, show that S = {[0], [a], [2a], · · · , [m − a]} forms a (possibly
    nonunitary) subring of Z/mZ when a divides m. (i.e. show that (S,+, ·) is closed
    the usual addition and multiplication. (We are not require to find a multiplicative identity).



    3. The attempt at a solution

    Since a divides m then m=ab so I tried subbing in ab for m and got [m-a]=[ab-a]=[a(1-b)]... but not too sure where to go from here. From looking at the set S it does not seem to be closed under addition or multiplication? Just a hint at how to go about/start/ approach this question would much appreciated! THank you!
     
  2. jcsd
  3. Feb 20, 2013 #2

    jbunniii

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    Well, let's check closure under addition. Two general elements of ##S## look like ##[ra]## and ##[sa]##, where ##r## and ##s## are integers. So what is ##[ra] + [sa]##? Evaluate it in the ring ##\mathbb{Z}/m\mathbb{Z}##, and see if the answer is in ##S##.
     
  4. Feb 21, 2013 #3
    OK, I see how to prove it now- the addition of two elements of S will always give a number which is a multiple of a, therefore will be an element of S. (Similarly for multiplication)...?

    (Thank you)
     
  5. Feb 21, 2013 #4

    jbunniii

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    Yes, similarly for multiplication.
     
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