(A) is contained in I if and only if A is contained in I

Let R be a commutative ring, let I \subseteq R be an ideal, and let A \subseteq R be a subset. Prove that (A) \subseteq I if and only if A \subseteq I.


Certainly if (A) \subseteq I then A \subseteq I. Suppose conversely that A \subseteq I. Now (A) consists of all finite R-linear combinations of elements from A. Since A \subseteq I and I is an ideal, (A) \subseteq I.

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