Suppose R is a ring with identity and J is the left ideal of R generated by $\{ab - ba: a, b \in R\}$. Does J always have to be a two-sided ideal?

I failed to construct the counterexample, but I do not know how to prove this statement either.

Any help will be appreciated.


1 Answer 1


It suffices to show that given any element $[a,b]=ab-ba$, the element $(ab-ba)r\in J$ for all $a,b,r\in R$.

We have $$\begin{align*}(ab-ba)r&=abr-bar\\&=abr-arb+arb-bar\\&=a[b,r]+[ar,b]\end{align*}$$ which is in $J$ because $J$ is a left ideal.


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