Why echo "R" > /dev/ttyUSB0 doesn't work, but on the same PC Serial Monitor in Arduino IDE works?
I had the same issue.
After entering command that Majenko suggested:
stty -F /dev/ttyUSB0 -hupcl

I still wasn't able to send commands to Arduino nano.
Strangely, when I open Arduino IDE Serial Monitor and send command it works.
When I close it, I can send commands in terminal to Arduino for some time. Then it stops working again :/