You have to go online once initially. You also get a 30 day trial of Lightroom mobile, so it may be that you've activated that, that that's what's going online.
I have registered in my account and paid for the program, do you mean I can now log out of the cloud and use the program
You are not logged into "the cloud" in the sense that you now have access to cloud features such as photo sharing. You do not have access to cloud features. You have activated your software. These are two different things.
Once you activate your software by signing in, you can now use Lightroom 6 without the 30-day restriction.
The question is: how do you get rid of this unwanted 'Lightroom mobile' test version?
I didn't activate it, it was immediately active after I first started 'Lightroom 6'. Since I didn't knew anything about it I looked into my account details. There I found an entry about a free trial abo, but I couldn't find any option to deactivate it.
Since I don't want Lightroom to send anything to any server I tried logging out inside Lightroom 6. But when I restarted the program it was started as 'Lightroom CC' test version instead. The system information menu told "expired trial license" instead of "unlimited license" like before. The only way I could figure out to get back 'Lightroom 6' with my unlimited license was to reinstall the program. But afterwards 'Lightroom mobile' was active again.
So I tried running it without internet connection to avoid any data transfer. It did work but every few minutes it got stuck for 10 to 20 seconds, which was really annoying. Only after I connected the computer to the internet this behavior stopped. This seems to be reproducible because it occurred again after the next program start without internet connection. And it stopped again after the connection was established.
Since I don't want to quit 'Lightroom 6' each time while my computer is connected to the internet for the next 29 days I would really like to know how to get rid of this really annoying forced 'Lightroom mobile' trial?
Turn it off in the Activity Center, which is there when you click the Identity Plate. Sync is the top item.
Turning the syncing off in the Activity Center (whatever this actually means) doesn't really deactivate the trial. And it also doesn't solve the problems that I must be logged in to Adobe and that Lightroom gets stuck every minute for 10 to 20 seconds if no internet connection is available.