Ring and elements with multiplicative decompositions with same sum

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