Every field is an integral domain?

[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?

 

[mfb]

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.

 

[HallsofIvy]

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?