Question:

Let R be a ring. Show that R = 0 = {0} if and only if 1 = 0 in R

How do you show it? I just know that 1 is the identity of multiplication group and 0 is the identity of addition group.

Thanks alot

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- October 12th 2009, 01:22 AMknguyen2005ring theory
Question:

Let R be a ring. Show that R = 0 = {0} if and only if 1 = 0 in R

How do you show it? I just know that 1 is the identity of multiplication group and 0 is the identity of addition group.

Thanks alot - October 12th 2009, 01:33 AMaman_cc
- October 12th 2009, 09:01 AMknguyen2005
Thanks for your reply, I understand the 1st part but I am not sure about last part

you said x.0 = 0.x = x for**all x in R**.

Can you explain to me how do you get this?

Thanks again - October 12th 2009, 09:09 AMaman_cc
Hi -

R = {0} so when I say**all x in R,**only value x can take is x=0. I said**for all x in R**just to make it exactly similar to how you define element 1 for any Ring.

if e.x = x.e = x for ALL x in R, then e is unique and is called multiplicative unity (1).

Now in our example R={0), put e=0

0.x = x.0 = x for all in x R.

For any other Ring this is true ONLY if x=0. But as in our special example x=0 is the only element in the Ring we can use for all x in R and hence the definition of 1 is satisfied, thus e=0=1.

I beleive this question is more to do with the understading of definitions than anything else.