SCHWEITZER 2010 Problem 4

Prove that if $n \geq 2$ and $I_ {1}, I_ {2}, \ldots, I_ {n}$ are idealized in a unit-element commutative ring such that any nonempty $H \subseteq \{ 1,2, \dots, n \}$ then if $\sum_ {h \in H} I_ {h}$ Is ideal $$I_ {2} I_ {3} I_ {4} \dots I_ {n} + I_ {1} I_ {3} I_ {4} \dots I_ {n} + \dots + I_ {1} I_ {2} \dots I_ {n-1}$$ also Is ideal.