Ring and elements with multiplicative decompositions with same sum

Question

Is there any commutative unitary ring $(R,+,\cdot,0,1)$, that is not an integral domain, with an element $x\neq 1,0$ that presents at least two distinct multiplicative "proper" decompositions (i.e. two couples $\{a,b\}\neq\{c,d\}$ such that $ab=x=cd$ and $a,b,c,d\neq 1$) for which the set $\{a+b\mid ab=x\land a,b\neq 1\}$ is a singleton? I'm looking for an example, but a proof of existence (or not existence) is enough.