If set of divisors of elements of a difference set $\Delta S = S - S \subset R$ a commutative ring is the whole ring (!) then $\Delta S$ is an ideal

Question

Let $R$ be an infinite commutative ring. Let $S \subset R$ be such that $\Delta S = \{ s - s' : s,s' \in S\}$ has the properties:

1. $$ \forall a \in R\setminus 0,\ a \textbf{ divides } g \textbf{ for } \infty\textbf{-many } g \in \Delta S $$

In other words the set of divisors of elements of $\Delta S$ is the whole ring! And each divisor occurs infinitely often!

Is there some way we can conclude that $\Delta S$ forms an ideal of $R$?

Answer 1

Consider $R=\mathbb{Z}$ and $S=\{n!: n\geq k\}$.