Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect to $S$, could we recover the ring structure of $R$?
In particular, if $X$ is a $H$-space, $R=H_*(X)$, $S=\pi_0(X)$, and we know the ring structure of $H_*(X)[\pi_0X^{-1}]$, can we obtain the ring structure of $H_*(X)$?