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?