Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

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

1 Answer 1

up vote 5 down vote accepted

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

share|improve this answer
1  
Maybe I should give a break :) That is so easy, I'm ashamed. –  user20353 Jan 4 '12 at 20:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.