On the sumsets of exceptional units in quaternion rings

We investigate sums of exceptional units in a quaternion ring H(R) over a finite commutative ring R. We prove that in order to find the number of representations of an element in H(R) as a sum of k exceptional units for some integer k ? 2, we can limit ourselves to studying the quaternion rings over local rings. For a local ring R of even order, we find the number of representations of an element of H(R) as a sum of k exceptional units for any integer k ? 2. For a local ring R of odd order, we find either the number or the bounds for the number of representations of an element of H(R) as a sum of 2 exceptional units.