An elementary proof of Gabrielʹs theorem on degenerate bilinear forms and its generalization

A theorem due to P. Gabriel shows that every bilinear space decomposes in a unique manner, up to equivalence, as the orthogonal direct sum of indecomposable degenerate spaces and a nondegenerate space. This has been generalized, by C. Riehm and M.A. Shrader-Frechette, to sesquilinear modules over a semisimple ring with an anti-automorphism. We give an elementary proof of these facts which avoids the use of Kronecker modules.