Exercise 4, § 5

Let M be a finitely generated projective A-module; for every bilinear mapping $u$ of $M \times M$ into M, let $\tilde{u}$ denote the tensor $\theta_M^{-1}(u)$ in $T_{\{1, 2\}}^{(3)}(M)$. Show that for the algebra structure on M defined by $u$ to be associative, it is necessary and sufficient that the tensors $c_4^3(\tilde{u} \otimes \tilde{u})$ and $c_2^6(\tilde{u} \otimes \tilde{u})$ correspond under the canonical associativity isomorphism of $T_{\{1, 2, 3\}}^{(4)}(M)$ onto $T_{\{1, 3, 4\}}^{(2)}(M)$.