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Math Help - [SOLVED] Rings

  1. #1
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    [SOLVED] Rings

    Hi, i'm finding the following question difficult;

    Let R be a ring with a one. Show that if 1=0 in R (i.e 1_R=0_R) then 0 is the only element in R , so that R is the zero ring.

    Also in Z_10 is it right that the units are 1, 3, 7 and 9 with inverses 1, 7, 3, 9, respectively?

    Thanks for your help!
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  2. #2
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    Quote Originally Posted by Louise View Post
    Hi, i'm finding the following question difficult;

    Let R be a ring with a one. Show that if 1=0 in R (i.e 1_R=0_R) then 0 is the only element in R , so that R is the zero ring.
    That's not difficult at all! In any ring ab= a(b+ 0)= ab+ a0 so a0= 0 for all a. But 1a= a for all a as well.

    Also in Z_10 is it right that the units are 1, 3, 7 and 9 with inverses 1, 7, 3, 9, respectively?
    Just do the calculations: 1*1= 1, 3*7= 21= 1, 7*3= 21= 1, 9*9= 81= 1 so those are units. 2, 4, 6, and 8 cannot be units because any multiple must be even. 5 cannot be a unit because any multiple must end in 0 or 5.

    Thanks for your help!
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