Answer:
If Iᵏ ⊆ J for some k ≥ 1, then radI ⊆ radJ.
Explanation:
To prove the statement (a), we need to show that if Iᵏ ⊆ J for some k ≥ 1, then radI ⊆ radJ, where radI and radJ represent the radical of the ideals I and J, respectively.
Recall that the radical of an ideal is defined as follows:
radI = r ∈ R
Now, let's proceed with the proof:
Suppose Iᵏ ⊆ J for some k ≥ 1.
We want to show that radI ⊆ radJ.
Take any element r ∈ radI. This means that rⁿ ∈ I for some n ≥ 1.
Since Iᵏ ⊆ J, we have rⁿ ∈ J for the same n.
Now, we need to show that r ∈ radJ.
To prove this, we need to show that rⁿ ∈ J for some n ≥ 1.
Since rⁿ ∈ J and J is an ideal, we can conclude that rⁿ ∈ J for some n ≥ 1, which means that r ∈ radJ.
Therefore, we have shown that if Iᵏ ⊆ J for some k ≥ 1, then radI ⊆ radJ.
This completes the proof.