On flexible quadratic algebras (Q949846)

Let \(R\) be a commutative associative ring with a unit element, and \(F\) a finitely generated projective \(R\)-modules of a constant rank \(s\). Suppose that \(F\) is provided with a nondegenerate symmetric \(R\)-bilinear form \(B:F\otimes_RF\to R\) and with an alternative \(R\)-bilinear map \(\times: F\otimes F\to F\) such that \[ B(x\times y,x\times y)=B(x,x)B(y,y)-B(x,y)^2,\quad B(x,x\times y)=0 \] for all \(x,y\in F\). Define a multiplication on \(C=R\oplus F\) by the rule \[ (r_1+f_1)(r_2+f_2)=\left(r_1r_2-B(f_1,f_2)\right)+\left(r_1f_2+r_2f_1+f_2\times f_1\right). \] Then \(F\) is a composition \(R\)-algebra and \(s=1,3,7\). Conversely, let \(C\) be a composition \(R\)-algebra with a norm \(n\) and a canonical involution \(x\mapsto\bar x\). Suppose that the rank of \(C\) is equal to \(s+1\). Then \(C=R\oplus F\) where \(F\) is the set of all skew-symmetric elements. Put \(x\times y=\frac 12 [x,y]\) and \(B(x,y)=\frac 12(xy+yx)=\frac 12 n(x,y)\). Then all previous conditions are satisfied and \(C\) is obtained by the construction presented above. These results are known for the case when \(R\) is a field [see \textit{K.~A.~Zhevlakov, A.~M.~Slin'ko, I.~P.~Shestakov} and \textit{A.~I.~Shirshov}, Rings that are nearly associative (in Russian). Moskva: Nauka (1978; Zbl 0445.17001), English translation: New York etc.: Academic Press (1982; Zbl 0487.17001)].