[Math] cancellation laws in a Ring


There's a theorem that states that cancellation laws hold in a ring $R$ if and only if $R$ has no zero divisors. Note that Integral Domains have no zero divisors. However, from my understanding in group theory, cancellation law happens by multiplying the (multiplicative) inverse on both sides, i.e. $$a^{-1}\cdot ab=a^{-1}\cdot ac\implies b=c.$$ Equivalently, $$ba\cdot a^{-1}=ca\cdot a^{-1}\implies b=c.$$ Going back to rings, this feels counterintuitive as this is only possible if all elements have a multiplicative inverse. But take note that Integral Domains are not necessarily division rings. So how does cancellation exactly work in rings?

Our professor said we should not multiply the inverse on both sides since we're not guaranteed that a multiplicative inverse exists for all elements of a ring $R$. So for the rest of the discussion, she only canceled terms without explicitly stating why it happens.

Best Answer

Suppose $ab=ac$. This means that


but if the ring is an integral domain, either $a=0$ or $b-c=0$. If $a\neq0$, this implies that $b=c$.

Related Question