## (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, $(A) \subseteq I$.