1. Introduction. Let 1 and 2[i] have their usual meaning. Let Y o denote the noncommutative ring of integral quaternions, that is the set of all elements a + bi + cj + dk with a,b,c,deZ and where i, j and k together with the number 1 are the four units of the system of quaternions.Let ^V((x) = a 2 + b 2 + c 2 + d 2 be the norm of the element jx = a + bi + cj + dke Y o . The nontrivial ideals in Y o are exactly the principal ideals (/x) generated by elements /x e Y o with M{{i)^2.Analogous to the definition of uniformly distributed sequences in Z due to Niven [4] (see also Kuipers and Niederreiter [1, Chapter 5]) and that in Z[i] due to Kuipers, Niederreiter and Shiue [2] we consider sequences of integral quaternions and ask how they are distributed modulo an arbitrary nontrivial left ideal in Y o . We introduce the following definition.
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