Let R be a ring with an identity 1, and R′ a ring anti - isomorphic to R. Let V be an R-module as well as an R′-module. We assume that 1 a = a for all elements a in V and that V satisfies the minimum condition for R-submodules. Elements of R will be denoted by α, β, …, and those of V by a, b, … Elements of R′ will be a α′, β′, …, where α′ fcorresponds to α by the anti - isomorphism. A mapping f of V x V to R is called a bilinear mapping of V to R if it satisfies the following.