It is proved that for adjointable operators A and B between Hilbert C*-modules, certain
majorization conditions are always equivalent without any assumptions on <(R(A*))over
bar>, where A* denotes the adjoint operator of A and <(R(A*))over bar> is the norm
closure of the range of A*. In the case that <(R(A*))over bar> is not orthogonally
complemented, it is proved that there always exists an adjointable operator B whose
range is contained in that of A, whereas the associated equation AX = B for adjointable
operators is unsolvable.