Why are liquid ring compressors not used in a jet engine? A question has been asked here before whether sliding vane compressors could be used in place of axial compressors on jet engines since they have a high compression ratio per stage (Why were rotary vane compressors never used in jet engines?), the answer being that the sliding seal would give out at the RPMs required.

What about a liquid ring pump?

It's essentially a sliding vane compressor where the seal is provided by the mass of liquid moving along the circumference. Is it possible to maintain a mass flow of the same order as axial compressors with these? Please ignore the question of cooling of the liquid for the purpose of this discussion.
You said to ignore the cooling of the liquid, but that's something that cannot be ignored.  With the compression ratios achieved in typical jet engine compressors, the air leaving the last stage of the compressor may be 800 - 1000 deg F.  You won't find a lot of liquids that won't just completely vaporize at those temperatures.  Yes you could potentially cool the liquid, but the cooling apparatus would add a huge amount of weight, as well as consume a lot of power.