>>13300114no u! Did you know you can EXPLODE a ring (or an algebra over a commutative ring) but still keep it homologically the same? If R is a commutative ring (integers and you get rings) and A is an R-algebra (differential graded, but you can assume it is all in degree 0 with zero differentials to get a "normal" algebra), then you can take the bar construction on A to get a DG R-coalgebra B(A). There is then the Adams cobar construction taking DG coalgebras to DG algebras, and applying that to B(A) gives you ?B(A). Those are actually adjoint functors, ? being the left adjoint, so there is the counit sending stuff from ?B(A) to A. This is actually a quasi-isomorphism, so those two have the same homologies. However, the bar construction on A is itself a huge thing, a free tensor coalgebra, and the cobar on a coalgebra is a free tensor algebra, so you end up getting a free tensor algebra on a free tensor coalgebra... That's gigantic!!! But even then it has the same homology as whatever you started with. With coefficients in R that is.
