On Rings of Continuous Functions on Topological Spaces

The present note deals with the same subject as the investigations by M. Stone (2) and the note of G. Šilov published above. In difference from this last note, we consider a ring of continuous functions defined on a rertain topological space as a purely algebraical formation, without introducing in it any topological relations. It turns out that in the case of bicompact spaces considered by M. Stone, as well as in considerably more general cases, the algebraical structure of the ring of continuous functions already defines the topological space up to a homeomorphism.