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.e. kill in $H^\ast(A)$ all sums of elements that can be obtained via $m_i$ taking elements of $S$ as one of the arguments)? Does this quotient possess any nice 'universal' properties? What other reasonable definitions of $H^\ast(A)/\langle S\rangle$ can one consider here?

I would like to concider the (triangulated?) category of $A_\infty$-modules over $H^\ast(A)/\langle S\rangle$; is there a way to describe it 'explicitly'? What is the relation between $A$-modules and $H^\ast(A)/\langle S\rangle$-ones? Can one define a functor by $A-\mod\cong H^\ast(A)-\mod \stackrel{\otimes_ {H^\ast(A)}H^\ast(A)/\langle S\rangle}{\to} H^\ast(A)/\langle S\rangle-mod$; does it possess any nice universal properties?

What are the 'canonical' references for these matters (in particular, for factorizing $A_\infty$-algebras modulo ideals)?

Upd. Unfortunately, I would like to consider an algebra that is not skew-commutative; it is closely related with a 'complicated' DG-category.