Let $A$ be a differential graded algebra, $S\subset H^*(A)$. I would like to 'kill $S$ in a canonical way'. Is it possible to do it as follows: consider the $A_\infty$-algebra structure on $H^\ast(A)$, and factorize $H^*(A)$ by the '$A_\infty$-ideal' generated by $S$? I would like to concider the (triangulated?) category of $A_\infty$-modules over the algebra obtained; is there a way to describe it 'explicitly' (in particular, the part of it that comes from the category of $A$-modules)?

What are the 'canonical' references for these matters?