We investigate the relationship between the determinants det P and norms NP of right modules P of rank one over associative composition algebras C. We show: det P NP NP if C is a quaternion algebra, and det P NP det C if C is a quadratic e  tale algebra.1. Introduction. The determinant of a module over a ring is a well-known and important invariant in module theory. In [3] Knus implicitly showed that the determinant of a quaternion composition algebra over an arbitrary (unital associative commutative) ring R is trivial, i.e., isomorphic to R. The proof of this fact consists of several steps. Firstly, one shows that the Arf invariant of a quaternion composition algebra Q over R is trivial, i.e., isomorphic to R  R, secondly, this implies the triviality of the discriminant module of Q, and thirdly, by method of descent, one obtains an isomorphism between the discriminant module and the determinant of Q.In this paper a very simple proof of the triviality of a quaternion composition algebras determinant is given.In [2] Knus introduced the concept of a quadratic space relative to a hamiltonian quaternion algebra over a real scheme. Such a quadratic space can be thought of as a quaternionic line bundle. In [3] we find the notion of norm forms on right modules over quadratic resp. quaternion algebras over arbitrary (unital commutative associative) rings. All these concepts have in common an associative composition algebra C with norm n C , a right C-module P, a quadratic form N P X P 3 L, where L is an invertible module such that N P pc n C cN P p is satisfied.Such structures occur quite naturally in H. P. Peterssons Cayley-Dickson doubling process (cf.[4]), where P is a right C-module, N P X P 3 R is a nonsingular quadratic form satisfying N P pc n C cN P p. This process gives rise to a composition algebra having the twice rank of C.S. Pumplu È n at University of Regensburg and H. P. Petersson at FernUniversita È t Hagen communicated to the author the question whether the determinant of P is isomorphic to L L or L. In this paper we investigate the relationship between the determinant of P and the module L in case N P X P 3 L is nonsingular, and answer this question positively.
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