Consider ordinary homology with coefficients in a field. For $X$ a path-connected pointed space, the graded vector space $\bigoplus\_{q\ge 0} H\_q(\Omega X)$ has the structure of an algebra with the multiplication induced as follows: $$ H\_p(\Omega X) \otimes H\_q(\Omega X) \rightarrowtail H\_{p+q}(\Omega X \times \Omega X) \to H\_{p+q}(\Omega X)$$
Here, the first map is a monomorphism by Kunneth's theorem. The second map is induced by the concatanetion of loops $\Omega X \times \Omega X \to \Omega X$.


Let two path-connected pointed spaces $X$ and $Y$ be given. Suppose that, for each $q\ge 0$, the two vector spaces $H\_q(\Omega X)$ and $H\_q(\Omega Y)$ have the same dimension. Hence $\bigoplus\_{q\ge 0} H\_q(\Omega X)$ and $\bigoplus\_{q\ge 0} H\_q(\Omega Y)$ are isomorphic as graded vector spaces. Is it necessary that $\bigoplus\_{q\ge 0} H\_q(\Omega X)$ and $\bigoplus\_{q\ge 0} H\_q(\Omega Y)$ are isomorphic as algebras?



Does a graded vector space isomorphism between the homology of two loop spaces imply the existence of an algebra isomorphism?
No.


Let $X = B\Bbb Z/4$ and $Y = B(\Bbb Z/2 \times \Bbb Z/2)$ be classifying spaces for the two groups of order four. Then, as loop spaces, $\Omega X$ and $\Omega Y$ are homotopy equivalent to the discrete spaces $\Bbb Z/4$ and $\Bbb Z/2 \times \Bbb Z/2$ respectively. 


Their rational homology groups are the same, because the underlying spaces $\Omega X$ and $\Omega Y$ are homotopy equivalent: both of them are homotopy equivalent to a discrete set with four points. However, $H\_\*(\Omega X)$ is the group algebra $\Bbb Q[\Bbb Z/4]$ and $H\_\* (\Omega Y)$ is the group algebra $\Bbb Q[\Bbb Z/2 \times \Bbb Z/2]$. Those are nonisomorphic rings: one has three prime ideals, and the other has four.


