= Non-zero-divisor
In a <commutative ring>, an element $a$ is a <non-zero-divisor> when multiplication by $a$ is <injective>: $ab=0$ implies $b=0$. Equivalently, $ab=ac$ permits cancellation to $b=c$, since $a(b-c)=0$. In $\mathbb Z/6\mathbb Z$, the class of $2$ fails this condition because $2\cdot3=0$, while the class of $5$ satisfies it because it is a <unit in a ring>.
For a <noncommutative ring>, multiplication on the left and on the right can have different null spaces. The implication $ab=0\Rightarrow b=0$ specifies the left-multiplication condition; the two-sided condition additionally requires $ba=0\Rightarrow b=0$. Stating the commutative setting or the side of multiplication prevents these distinct notions from being conflated.
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