Afr. Mat. (2012) 23:53–84 DOI 10.1007/s13370-011-0018-x Generalized quadratic modules Jacques Helmstetter · Artibano Micali · Philippe Revoy Received: 14 October 2010 / Accepted: 2 February 2011 / Published online: 4 March 2011 © African Mathematical Union and Springer-Verlag 2011 Abstract In 1973 the japanese mathematician Kanzaki defined two categories of gener- alized 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 x 2 = 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. Keywords Quadratic forms · Clifford algebras · Witt rings · Fields of characteristic 2 Mathematics Subject Classification (2010) 15A63 · 15A66 · 11E81 1 Introduction Let K be a commutative, associative and unital ring; in Sects. 3, 4, 6, 7, 8, 9, this ring K is assumed to be a field of characteristic 2. The category Quad( K ) of quadratic modules over K is well known: its objects ( M, q ) are modules M over K provided with a quadratic form q : M → K , and its morphisms ϕ : ( M, q ) → ( M ′ , q ′ ) are the linear mappings ϕ : M → M ′ J. Helmstetter (B ) Institut Fourier, B.P. 74, 38400 Saint-Martin d’Hères, France e-mail: Jacques.Helmstetter@ujf-grenoble.fr A. Micali · P. Revoy Université Montpellier II, 34095 Montpellier, France e-mail: micali@math.univ-montp2.fr; amicali@orange.fr P. Revoy e-mail: revoyph@gmail.com 123