Statement
Left or right zero divisors in ring can never be units.
is proved in Wikipedia this way:
If $a$ is invertible and $ab = 0$
$0 = a^{−1}0 = a^{−1}ab = b$
I'm confused by third transition.
I suppose that more detailed version of proof written this way
$0 = a^{−1}0 = a^{−1}(ab) = (a^{−1}a)b = b$
shows us that ring is required to be associative for us to prove the statement.
Is it true?
I also need to note that the whole question occured because of my math book, where ring by definition is not necessarily associative under multiplication.