sum of two elements in a ring Suppose I have a commutative ring $R$ with unity.
And suppose we have three elements $a, b, c \in R$
such that $a+b = c$.
Is it possible that there are two units $\epsilon, \epsilon'$ (not both $1$)
such that $\epsilon a + \epsilon'b = c$?
Thanks!

It's not possible for every ring.
In $\Bbb F_2$, $1+0=1$, but you cannot find any units $\epsilon,\epsilon'$ with the properties you want.
More generally in any Boolean ring (with identity), and any $a+b=c$, you still can't accomplish the task, because the only unit available is the identity in a Boolean ring.
