I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+b=t$ is invertible then how can one verify the existence of certain $c$ and $c'$ in $R$ such that $ca+bc'=1$ (and so, $ca$ and $bc'$ are idempotents)?

It appears that this implication does not hold unconditionally; yet I would be deeply grateful for any hints that would allow to study it (note however that the case where $t-1$ is nilpotent is certainly simple). In this case I am interested in there is an extra conservativity assumption; in particular, in the quotient of $R$ by the two-sided ideal generated by $t-1$ non-invertible elements of $R$ do not become invertible. I have tried to relate my question with von Neumann regularity of elements of rings, but was not able to do this.