Answer to Question #106455 in Abstract Algebra for Garima Ahlawat

Question

Let R be a ring for which ab = ca implies b = c for all a,b,c belongs to R, a not equal to zero. Show that R is commutative.

Answer 1

Say $a,b$ are two elements of the ring. Let us denote $ab=c, ba=d$ , belonging to the ring, since it is closed under multiplication.

Now, we have $aba=aba\implies ca=ad\implies c=d$ , from the property of the ring.

Thus, $ab=ba$ for arbitrary $a,b$ in the ring. Hence, the ring is commutative.