# [SOLVED] Rings

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• Feb 15th 2009, 01:26 AM
Louise
[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!
• Feb 15th 2009, 02:51 AM
HallsofIvy
Quote:

Originally Posted by Louise
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.

Quote:

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.

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Thanks for your help!