Let $R$ be commutative ring with identity, $I$ an ideal of $R$, and $S$ a subset of $R$.
Under what conditions is the set $A=\{a+s\mid a\in I , s\in S \}$:
1- an ideal of $R$?
2- a prime ideal of $R$?
Of course one trivial candidate is the case where either $I$ or $S$ is a prime ideal containing the other.
Thank you for any help.