Toward a generalization of strong approximation theorem to a general PF field

We aim to generalize Eichler's strong approximation theorem, which is known for a division algebra D over a global field K, to the case that K is a general PF field. First we show by an example that the generalized theorem is false for SL1 (D). But if we replace SL1 (D) by the commutator group [D-x, D-x], the generalizaion may be possible. Though its validity is not yet known, in this paper we decompose the generalized theorem into two parts, one of which can be formulated in a more general case that K is the quotient field of a Dedekind domain. Further, we prove the equivalence of four approximaion properties (a) similar to (a'''), which was open in our previous paper.