Geodesic
Structure Theorems of a Module Over a Ring with a Bilinear Mapping
Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 649-652

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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. 
Nobusawa, Nobuo. Structure Theorems of a Module Over a Ring with a Bilinear Mapping. Canadian mathematical bulletin, Tome 10 (1967) no. 5, pp. 649-652. doi: 10.4153/CMB-1967-063-5
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     author = {Nobusawa, Nobuo},
     title = {Structure {Theorems} of a {Module} {Over} a {Ring} with a {Bilinear} {Mapping}},
     journal = {Canadian mathematical bulletin},
     pages = {649--652},
     year = {1967},
     volume = {10},
     number = {5},
     doi = {10.4153/CMB-1967-063-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1967-063-5/}
}
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