Generalized quadratic modules

In 1973 the japanese mathematician Kanzaki defined two categories of generalized quadratic modules over every commutative, associative and unital ring K. In both categories a generalized quadratic module (M, f, q) is provided with a quadratic form q and a linear form f. These categories give something new only when 2 is not invertible in K, and here they are studied when K is a field of characteristic 2. The first category is fit for the definition of generalized Clifford algebras Cℓ(M, f, q) where x2 = f(x)x + q(x) for all x∈ M; all resulting Clifford algebras are described here. The second category is fit for the definition of nondegenerate objects, metabolic objects, orthogonal sums of objects, tensor products and extended Witt rings; we have managed to bring much information on the extended Witt ring We(K), and to propose an application. © 2011 African Mathematical Union and Springer-Verlag.