  Elements of a Vahlen group are $2 \times 2$ matrices with entries in a
Clifford algebra satisfying some conditions. Traditionally they have come in
both ordinary and paravector type and have been defined (over Clifford
algebras) over the real or complex numbers. We extend the definition of both
types to be over a commutative ring with an arbitrary quadratic form. We show
that they are indeed groups and identify in each case the group as the pin
group, spin group, or another subgroup of the Clifford group. Under some mild
conditions, for both types we show the equivalence of our definition with a
suitably generalised version of the two standard definitions.
