MAPS DETERMINED BY ACTION ON SQUARE-ZERO ELEMENTS

Let M-n(R) be the algebra of all n x n matrices over a unital commutative ring R with 6 invertible, and let X be an R-module. It is shown that if a symmetric bilinear map {.,.} : M-n(R) xM(n)(R) -> X satisfies the condition that {u, u} = 0 whenever u(2) = 0, then there exists a linear map f : [sl(n)(R)](2) -> X such that {x, y} = f(x circle y), for all x, y is an element of sl(n)(R), where sl(n)(R) denotes the subset of M-n(R) of all matrices of trace zero.