How can the ideal generated by ab-ba force a ring to commute?

[markiv]

Is there something you can do to a ring to produce a commutative ring? Like for any group, you can create an Abelian group by factoring out its commutator subgroup. Can you "force" a ring to commute?

 

[Tinyboss]

For a group, modding out by the commutator subgroup gives the largest abelian quotient. You could make a ring commutative by simply redefining multiplication such that cd=0 for every c,d in the ring, but I think what you're asking for is a "least destructive" way of making a ring commutative. I don't know the answer.

 

[arkajad]

The simplest idea that comes to my mind: What about factoring the ring by the two-sided ideal generated by ab+ab?

 

[Landau]

I think you mean the ideal generated by ab-ba?

 

[mathwonk]

the most interesting rings are the matrix rings. you might think about how destructive it would be to kill all elements of form AB-BA.

also for a group you might reflect on the difference between the free group on two generators and the free abelian group ZxZ.

 

[arkajad]

I think you mean the ideal generated by ab-ba?

Yes. My typo.