Proving Intersection of Ideals is an Ideal

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SUMMARY

The intersection of any set of ideals in a ring is itself an ideal. This is established by applying the definitions of an ideal, specifically the properties that for any elements \( a \) and \( b \) in the intersection, \( a - b \) must also be in the intersection, and for any element \( r \) in the ring, both \( ra \) and \( ar \) must be in the intersection. The proof involves taking two elements from the intersection and demonstrating that they satisfy the ideal properties directly.

PREREQUISITES
  • Understanding of ring theory and the definition of a ring.
  • Familiarity with the concept of ideals in algebra.
  • Knowledge of set theory, particularly intersections of sets.
  • Basic proficiency in mathematical proofs and symbolic representation.
NEXT STEPS
  • Study the properties of ideals in ring theory.
  • Learn about the different types of ideals, such as maximal and prime ideals.
  • Explore examples of rings and their ideals, focusing on polynomial rings.
  • Investigate the role of intersections in other algebraic structures.
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Mathematics students, algebra enthusiasts, and educators looking to deepen their understanding of ring theory and the properties of ideals.

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Homework Statement


Prove that the intersection of any set of ideals of a ring is an ideal.


Homework Equations


A nonempty subset A of a ring R is an ideal of R if:
1. a - b ε A whenever a, b ε A
2. ra and ar are in A whenever a ε A and r ε R


The Attempt at a Solution


My guess is that i need to start with a collection of ideals,
write a representation of the form of the intersection of those ideals,
upon which i can take two generic elements and apply the ideal test above

Putting this into symbols seems to be the tricky part for me.
Thanks.
 
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You don't need a representation of the form of the intersection. Just apply the definition directly. For example, to apply 1, take a & b in the intersection. What can you say about a-b?
 

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