A Ring is Commutative if Whenever $ab=ca$, then $b=c$

Problems and solutions of ring theory in abstract algebra

Problem 615

Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$.

Then prove that $R$ is a commutative ring.

 
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Proof.

Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$.
Consider the identity
\[y(xy)=(yx)y.\] This can be written as $ab=ca$ if we put $a=y, b=xy, c=yx$.

It follows from the assumption that we have $b=c$.
Equivalently, we have $xy=yx$.

As this is true for any $x, y\in R$, we conclude that $R$ is a commutative ring.


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