Thank fedja for constructing a brilliant counterexample. I write it as follows.
Define R to be a class of finite unions of half-open rectangles in . It can be verified that R is a ring. Consider point set in . A can be expressed as a union where sequence is an indexed class of the set being unioned. First can not be reduced to a countable sequence with the same union, for otherwise the line would be obtained by a countable union of points, which contradicts the fact that any line is a continuum. Second, If uncountable sequence can be disjointliazed to be a uncountable sequence in R satisfying and disjoint, since every term , as an element of R, contains a point of rational coordinate, it would follow that there are uncountable distinct rational coordinates. However, the set of all points of rational coordinate in has countable cardinality. So, the disjointlization for sequence is not possible. The same argument applies when we are to extend this impossibility to -ring.