# Every field is an integral domain?

• PsychonautQQ
In summary, the conversation discusses the relationship between elements r and s in R, where rs = 0. The speaker is trying to prove that both r and s must be zero, but realizes that the logic only proves that if one of the elements is not zero, then the other must be zero. The mistake lies in assuming that both r and s have multiplicative inverses.

#### PsychonautQQ

Okay, so if r,s are elements of R and rs = 0 then either r or s has to equal zero.

I'm just confused because it seems that if rs = 0 then we can show that both r and s must be zero... I don't know what I'm doing wrong..

rs = 0
r^-1(rs) = r^-1(0) = 0
1s = 0
s = 0

and then:

rs = 0
r(ss^-1) = (0)s^-1 = 0
r1 = 0 = 0
r = 0

I mean, both r and s have multiplicative inverses so it seems like r and s both need to be zero by this logic.. what am i doing wrong here?

PsychonautQQ said:
I mean, both r and s have multiplicative inverses
Not if they are zero.

5*0=0 in the real numbers
Now try to apply your prove to show 5=0. It won't work.

PsychonautQQ said:
Okay, so if r,s are elements of R and rs = 0 then either r or s has to equal zero.

I'm just confused because it seems that if rs = 0 then we can show that both r and s must be zero... I don't know what I'm doing wrong..

rs = 0
r^-1(rs) = r^-1(0) = 0
You are assuming, when you write this, that r has an inverse. Which is only true if r is not 0.
1s = 0
s = 0
So what you have proved is "if r is not 0 then s is 0"

and then:

rs = 0
r(ss^-1) = (0)s^-1 = 0
r1 = 0 = 0
r = 0
again, you have proved that "if s is not 0 then r is 0".

I mean, both r and s have multiplicative inverses
no, this is not true.
so it seems like r and s both need to be zero by this logic.. what am i doing wrong here?

• PsychonautQQ