Explanation Given information: { I λ } , λ ∈ l , is an arbitrary collection of ideals I λ of the ring R . Formula used: Definition of ideal: A subset I of a ring R is an ideal of R , provided the following conditions are satisfied: 1. I is non-empty. 2. x ∈ I and y ∈ I implies x + y ∈ I . 3. x ∈ I implies − x ∈ I . 4. x ∈ I and r ∈ R implies x r and r x are in I . Proof: Let { I λ | λ ∈ l } be an arbitrary collection of ideals I λ of the ring R , where l is an index set such that for all λ ∈ l , I λ is an ideal of R . We have to prove that ∩ λ ∈ l I λ is an ideal of R , that is, to check the conditions for ideals are satisfied. 1. ∩ λ ∈ l I λ is non-empty: ∵ each I λ is an ideal of R ⇒ 0 ∈ I λ , ∀ λ ∈ l ∴ 0 ∈ ∩ λ ∈ l I λ Therefore, ∩ λ ∈ l I λ is non-empty. 2. x ∈ ∩ λ ∈ l I λ and y ∈ ∩ λ ∈ l I λ implies x + y ∈ ∩ λ ∈ l I λ : Let x ∈ ∩ λ ∈ l I λ and y ∈ ∩ λ ∈ l I λ ⇒ x , y ∈ I λ , ∀ λ ∵ each I λ is an ideal of R , and x , y ∈ I λ ⇒ x + y ∈ I λ , ∀ λ ⇒ x + y ∈ ∩ λ ∈ l I λ 3

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Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 6.1, Problem 5E

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To prove: ∩λ∈lIλ is an ideal of R.

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Chapter 6.1, Problem 4E

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To prove: ∩ λ ∈ l I λ is an ideal of R .

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