Introduction. Let S= (S, -) be a semigroup and T= QT(S, E) a semigroup of right quotients of S with respect to a subsemigroup E of S (cf. ?2). Suppose that S is equipped with a topology S which makes (S, ., C) into a topological semigroup. The purpose of this paper is to investigate topologies Z on T with the properties that (T, *, Z) is a topological semigroup and that the relative topology SZIS obeys IS= S or more generally IS c-There is a related question if one assumes that there exists even a group T= Qr(S, S) of right quotients and considers topologies Z on T which make (T, ., Z) into a topological group. Several authors have dealt with this topic [9], [8], especially in the commutative case [2], [4], [6], [7]. We shall speak later about some of their results, for which we obtain generalizations. The first part of this paper starts with the following question: Take (S, , ) and T= Qr(S, E) as above. Does there exist a topology Z on T, such that (T, , Z) is a topological semigroup, ZIS = C, and S is i-open ? We obtain (cf. ?3, Theorem 1) that this is the case if, and only if, all left and right translations by elements a. E E are open mappings in (S, *, (). Moreover, there is only one topology Z of this kind, and Z can be defined by the base