# Left ideal generated by $\lbrace ab-ba:a,b \in R \rbrace$ is a two-sided ideal

Let $R$ be a ring with $1$, and let $J$ be the left ideal of $R$ generated by $\lbrace ab-ba:a,b \in R \rbrace$. Then I want to show that $J$ is a two-sided ideal.

I thought that since $J$ is a left ideal, for any $r \in R$, $r(ab-ba)=rab-rba$ is in $J$ and I tried to show that $abr-bar$ is in $J$ but I failed. How should I continue?

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$(ab-ba)r = abr - bar = (a[br] - [br]a) + b(ra - ar)$ – Joel Cohen Jan 4 '12 at 20:56

Hint: $(ab-ba)r = a(br-rb) + ((ar)b-b(ar))$