Okay, so if r,s are elements of R and rs = 0 then either r or s has to equal zero.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Every field is an integral domain?

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