1. ## ring 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

2. Originally Posted by knguyen2005
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
IF 1=0
Consider any x in R. x=1.x=0.x=0
Thus R = {0}

If R={0} not sure if there is anything to prove. x.0 = 0.x = x for all x in R. Thus by definition 1 = 0.

3. 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

4. Originally Posted by knguyen2005
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
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.