Property of a ring with unity
Suppose $R$ be a ring with unity $I$ and $a$ belongs to $R$. How to prove that the ring is commutative if there exists a unique $b$ in $R$ such that $ab=I$?
My teacher told me to prove $ba=I$, consider the product $a(ba-I+b)$. But I just want know if there is any another way to prove that the ring is commutative.
Please help.
$\endgroup$ 61 Answer
$\begingroup$It is true that if $a$ is an element of a unital ring $R$ and there exists a unique $b\in R$ such that $ab= I$, then also $ba=I$. In this case, $b=a^{-1}$, and $b$ commutes with $a$, but this does not imply that $a$ commutes with everything else, let alone that the ring is commutative. As Mariano mentioned, there are noncommutative rings in which every element is invertible, and as tetori mentioned, the ring of quaternions gives a specific counterexample.
Your teacher gave you a good hint for showing that $ba=I$. First you can simplify $a(ba-I+b)$, then you can use uniqueness of $b$ to finish.
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