This is a question i asked on math.stackexchange but i didn't get any answer.

Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we have obvious lift of multiplication $f:A\otimes A\rightarrow A$ to $F:P\otimes P \rightarrow P$. Of course there is no reason for $F$ to be associative so we can't claim that $P_{\bullet}$ has algebra structure. Consider the map

$Tor_*(A,A)\otimes Tor_*(A,A)=H_*(P_{\bullet}\otimes P_{\bullet})\otimes H_*(P_{\bullet}\otimes P_{\bullet})\rightarrow H_*(P_{\bullet}\otimes P_{\bullet}\otimes P_{\bullet}\otimes P_{\bullet})\rightarrow H_*(P_{\bullet}\otimes P_{\bullet})=Tor_*(A,A)$

I was wondering if this map gives graded algebra structure on $Tor_*(A,A)$. Do i need any assumptions?

Edit: Can you give me references to any papers about those algebras?