The answer to this question is no in general.  Let $K$ be a field, and let $R=F\{s,t\ :\ st=s+t \}$, the non-unital algebra generated by the non-commuting variables $s,t$ subject to the single relation $st=s+t$.

Let $A$ be *any* unital $K$-algebra containing $R$.  In $A$ we have $(1-s)(1-t)=1-s-t+st=1$.  However, $(1-t)(1-s)=1-t-s+ts\neq 1$ since $ts\neq s+t$.  Thus $(1-s)$ is only right invertible, and hence even if $R$ is an ideal of $A$ we know that $A/R$ is not a division ring.