 
Summary: NONASSOCIATIVE QUATERNION ALGEBRAS OVER
RINGS
S. PUMPL¨UN, V. ASTIER
Abstract. Nonsplit nonassociative quaternion algebras over fields were
first discovered over the real numbers independently by Dickson and Al
bert. They were later classified over arbitrary fields by Waterhouse.
These algebras naturally appeared as the most interesting case in the
classification of the fourdimensional nonassociative algebras which con
tain a separable field extension of the base field in their nucleus. We
investigate algebras of constant rank 4 over an arbitrary ring R which
contain a quadratic ´etale subalgebra S over R in their nucleus. A gen
eralized CayleyDickson doubling process is introduced to construct a
special class of these algebras.
Introduction
Let k be a field. A nonsplit nonassociative quaternion algebra over k
is a fourdimensional unital kalgebra A whose nucleus is a separable qua
dratic field extension of k. Nonsplit nonassociative quaternion algebras
were early examples of nonassociative division algebras which are neither
powerassociative nor quadratic and were first considered by Dickson [D] in
1935, and by Albert [A] in 1942, both times over the reals. In 1987, Water
