With the ATNF Mopra telescope we are performing a survey in the ¹²CO(1–0) line to map the molecular gas in the Large Magellanic Cloud. For some regions we also obtained interferometric maps of the high density gas tracers HCO⁺ and HCN with the Australia Telescope Compact Array. Here we discuss the properties of the elongated molecular complex that stretches about 2 kpc southward from 30 Doradus. Our data suggest that the complex, which we refer to as the ‘molecular ridge’, is not a coherent feature but consists of many smaller clumps that share the same formation history. Likely triggers of molecular-cloud formation are shocks and shearing forces that are present in the surrounding south-eastern Hi overdensity region, a region influenced by strong ram pressure and tidal forces. The molecular ridge is at the western edge of the the overdensity region where a bifurcated velocity structure transitions into a single disk velocity component. We find that the ¹²CO(1–0) and Hi emission peaks in the molecular ridge are typically near each other but never coincide. A likely explanation is the conversion of warmer, low-opacity Hi to colder, high-opacity Hi from which H₂ subsequently forms. On smaller scales we find that very dense molecular gas, as traced by interferometric HCO⁺ and HCN maps, is associated with star formation along shocked filaments and with rims of expanding shell-like structures, both created by feedback from massive stars.