the best inexpensive water cooling with one fan and very quiet for the processor?
What CPU and case do you have or are planning to get? Those 120 or 140mm AIO coolers are not very good cost to performance and quite often a budget tower cooler can outperform them and cost much less. A well designed air cooler with a silent fan will make less noise than any AIO out there. Sure an AIO can use the same silent fan on the radiator but it will still have a pump that makes noise that an air cooler won't have.
The best won't be the cheapest, you don't get a AIO for silence, and 120MM/140MM AIO's are of poor value.
As other stated, a 120mm AIO is a poor value, considering the cost vs the performance output.
For the same amount of money (if you insist on an AIO), just get this one.
It costs reasonable and has a larger radiator than a 120mm AIO.
Now as far as the noise concerns, the most AIOs are pretty noisy at full speed.
And how you will configure an AIO, depends from the processor and if you wanna OC or not.