On iOS you can do this by pressing the status
indicator in the top right (the coloured square) and turning off "Accept
push notifications".
On Android you can do this by pressing the hamburger menu in the top left and toggling "Silent" (the bell icon).
Whilst both of the above settings will still technically make the mobile app 'ring', it won't make a noise or notify you. If you don't want the app to ring at all, the best alternative is to fully sign out of the app.