Let $A$ be a commutative ring of characteristic zero. Let $a_1,a_2,a_3,a_4 \in A$ be units such that $a_i^k\ne a_j^k$ for $i\ne j$ and $k=1,2$.
How to show that $a_1 a_2 a_3 + a_1 a_2 a_4 + a_1 a_3 a_4 + a_2 a_3 a_4 \ne 0$.
With a positive answer for this question, is that possible to generalize for $n$ units?
Remark: If necessary, $A$ could be considered a discrete valuation ring.