A commutative ring with unity and Ideals

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SUMMARY

This discussion focuses on the properties of ideals in a commutative ring with unity, specifically addressing the condition that if the sum of two ideals I and J equals the ring R (I + J = R), then their intersection equals the product of the ideals (I ∩ J = IJ). The participant confirms understanding that the product of the ideals is contained within their intersection (IJ ⊆ I ∩ J) but expresses difficulty in proving the converse. The discussion concludes without requiring further answers, indicating a resolution of the participant's inquiry.

PREREQUISITES
  • Understanding of commutative rings with unity
  • Familiarity with the concepts of ideals in ring theory
  • Knowledge of set operations, specifically intersection and union
  • Basic grasp of algebraic structures and their properties
NEXT STEPS
  • Study the proof of the property I + J = R implies I ∩ J = IJ in detail
  • Explore examples of commutative rings and their ideals
  • Learn about the role of unity in ring theory
  • Investigate other properties of ideals, such as maximal and prime ideals
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in abstract algebra, students studying ring theory, and anyone interested in the properties of commutative rings and ideals.

emptyboat
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Let R be a commutative ring with unity. I and J are ideals of R.
Show that If I + J = R, then I∩J=IJ.

I know that IJ⊆(I∩J).
But I can't do inverse.
 
Last edited:
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I solve it!
Thus no answer is required. :)
 

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