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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.

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    @Marc van Leeuwen: my book defines the ring without identity, associativity and commutativity under multi$p$lication. Just abelian group under addition, having both distributive laws and multiplication defined on set. Sorry for the mess, anyway. I never thought the math definitions from different books can be so... different.2011-11-30

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The sedenions form a nonassociative ring with identity in which every nonzero element has a multiplicative inverse, but there are zero divisors.