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Abstract

It is well known that for the ring ${\rm H}({\mathbb Z})$ of integral quaternions the unit group ${\rm U}( {\rm H}({\mathbb Z})$ is finite. On the other hand, for the rational quaternion algebra $ {\rm H}({\mathbb Q})$, its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ${\mathbb Z}\subset A \subseteq {\mathbb Q}$ such that the group of units ${\rm U}( {{\rm H}(A)})$ of quaternions over $A$ contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments as simple as possible.