In this paper topologies for the
quotient field K of a commutative inSegral domain A are investigated. The topologies
for K are defined so that convergence in K is stronger than convergence in A
whenever A is a topological ring. In particular, the Mikusinski field of operators is
the quotient field of many commutative integral domains which are also topological
rings. Each of these rings leads to a topological convergence notion in the Mikusinski
field, which is stronger than the convergence notion introduced originally by
Mikusinski. (The latter has recently been shown to be nontopological.) In general,
the algebraic and topological structures considered are not necessarily compatible;
however, the question of compatibility is investigated. Necessary and sufficient
conditions are given for the topology on A to be the restriction to A of the topology
defined on K. In a theorem of S. Warner, necessary and sufficient conditions have
been given for the neighborhood filter of zero in A to be a fundamental
system of neighborhoods of zero for a topology on K. Moreover K, with this
topology, is a topological field with A topologically embedded in K as an open
set. For rings satisfying the conditions of this theorem, the topology for
K which is defined in this paper is shown to reduce to that specified by
Warner.
