On topological extensions of Archimedean and non-Archimedean rings

The usage of the fields of p-adic numbers Q _p , rings of m-adic numbers Q _m and more general ultrametric rings in theoretical physics induced the interest to topological-algebraic studies on topological extensions of rational and real numbers and more generally (commutative and even noncommutative) rings. It is especially interesting to investigate a possibility to proceed to non-Archimedean rings by starting with real numbers. In particular, in this note we present “no-go” theorems (Theorems 3, 4) by which one cannot obtain an ultrametric ring by extending (in a natural way) the ring of real numbers. This puremathematical result has some interest for non-Archimedean physics: to explore ultrametricity one has to give up with the real numbers — to work with rings of e.g. m-adic numbers (where m > 1 is a natural, may be nonprime, number).