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

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SUMMARY

The discussion centers on methods to transform a non-commutative ring into a commutative ring, specifically through the ideal generated by the element ab - ba. Participants explore the concept of creating a commutative ring by factoring out the two-sided ideal generated by this element, drawing parallels to the formation of Abelian groups from non-Abelian groups by modding out the commutator subgroup. The conversation highlights the potential destructiveness of such transformations, particularly in the context of matrix rings.

PREREQUISITES
  • Understanding of ring theory and non-commutative rings
  • Familiarity with ideals in ring theory
  • Knowledge of matrix rings and their properties
  • Basic concepts of group theory, particularly Abelian and non-Abelian groups
NEXT STEPS
  • Research the properties of two-sided ideals in ring theory
  • Study the process of forming commutative rings from non-commutative rings
  • Examine the structure and properties of matrix rings
  • Learn about the relationship between groups and their Abelian quotients
USEFUL FOR

Mathematicians, particularly those specializing in abstract algebra, ring theorists, and anyone interested in the transformation of algebraic structures from non-commutative to commutative forms.

markiv
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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?
 
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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.
 
The simplest idea that comes to my mind: What about factoring the ring by the two-sided ideal generated by ab+ab?
 
I think you mean the ideal generated by ab-ba?
 
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.
 
Landau said:
I think you mean the ideal generated by ab-ba?

Yes. My typo.
 

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