Proving Intersection of Ideals is an Ideal

In summary, to prove that the intersection of any set of ideals of a ring is an ideal, we can apply the definition of an ideal by showing that the intersection satisfies the two conditions: 1) for any a and b in the intersection, a-b is also in the intersection, and 2) for any a in the intersection and r in the ring, both ra and ar are in the intersection.
  • #1
cubixguy77
8
0

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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  • #2
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?
 

Related to Proving Intersection of Ideals is an Ideal

What is the definition of an ideal?

An ideal in abstract algebra is a subset of a ring that satisfies certain properties, such as being closed under addition and multiplication by elements of the ring.

What does it mean to prove the intersection of ideals is an ideal?

Proving the intersection of ideals is an ideal means showing that the intersection of any two ideals in a ring is also an ideal in that ring.

Why is it important to prove that the intersection of ideals is an ideal?

Proving this property is important because it allows us to understand the structure of a ring better and make mathematical arguments about it. It also helps us solve problems and prove other theorems.

What are the steps to proving the intersection of ideals is an ideal?

To prove the intersection of ideals is an ideal, we need to show that it satisfies the definition of an ideal. This includes showing that it is closed under addition and multiplication by elements in the ring, and that it contains the additive identity and is closed under negation.

Can you provide an example of proving the intersection of ideals is an ideal?

For example, let I and J be ideals in a ring R. To prove that the intersection of I and J, denoted as I ∩ J, is an ideal in R, we need to show that for any x, y ∈ I ∩ J and any r ∈ R, we have (x + y) ∈ I ∩ J and (rx) ∈ I ∩ J. We also need to show that 0 ∈ I ∩ J and that for any x ∈ I ∩ J, we have -x ∈ I ∩ J. By showing these properties hold, we can conclude that the intersection of I and J is an ideal in R.

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