If $A$ is a commutative unital ring, $I$ is an ideal of $A$, and $a \in A$, then I know that $$aI=(a)I$$ where $aI=\{ax : x \in I\}$ and $(a)I$ is the ideal product of the principal ideal $(a)$ and $I$.
Is there a generalization of this for a subset $S\subseteq A$?
Many thanks!
Thanks to Mike Miller, I now know what it is I want to prove...
Theorem. If $A$ is a commutative unital ring, $I$ is an ideal of $A$, and $S\subseteq A$, then
$$(S)I=(SI)$$
where $SI=\{si:s\in S, i \in I\}$.
Proof. The ideal $(S)I$ is, by definition, the smallest ideal of $A$ containing $\{xi :x \in (S), i \in I\}$.
Hence, in particular, $SI\subseteq (S)I$ and so $(SI)\subseteq (S)I$.
Conversely, if $\sum_kx_ki_k\in (S)I$, then, as $(SI)$ is an ideal, $\sum_kx_ki_k\in (SI)$.
This completes the proof.