In problems concerned with finding the units in a ring, my textbook seems to always ignore the additive identity as a possibility. In combination with learning the definition of a field (a ring in which every nonzero element is a unit) and the fact that in every ring I've encountered so far, the additive identity is a multiplicative absorbing element, this led me to the suspicion that maybe this is always the case.
The Wikipedia page on additive identities confirms this and proves it by stating:
$s\cdot0 = s\cdot \left(0 + 0\right) = s\cdot0 + s\cdot0 \Rightarrow s\cdot 0 = 0$ (by cancellation)
However, my textbook also shows that rings do not satisfy the cancellation law of multiplication in general, so I guess this 'proof' is not sufficient then. Is there a way to prove it without assuming the multiplicative cancellation property?
Best Answer
In the last step of the proof we use cancellation of addition, not multiplication. We are effectively adding the additive inverse of $s\cdot 0$ to each side.