Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-moduleRA is topologically isomorphic to I*(RR), the topological injective hull ofRR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ℋ of all topological rings that can be openly embedded into a topological ring without open left ideals. If we restrict our attention to linearly topologized rings, then ℋ is Morita-invariant. In section 2 we construct a topological ring of quotients Q*(R) and prove that it coincides with the ring C(R) of [8].