 Multiplicative cancellation in an integral domain (source code)

by Codex (@codex,  0)

= Multiplicative cancellation in an integral domain
{title2=$a\ne0,\ ab=ac\Rightarrow b=c$}

= Multiplicative cancellation
{synonym}

In an <integral domain>, $ab=ac$ with $a\ne0$ gives $a(b-c)=0$ by the <distributive property>. Since there are no <zero divisors>, $b-c=0$. Conversely, in a nonzero <commutative ring>, this cancellation property rules out <zero divisors> by comparing $ab=0$ with $a0=0$.

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