There are?
I thought that in any case, if the virus decides to release itself to the world it kills the cell. I am serious suprised that it isn't true. Does it have a special name when it doesn't kill the cell?
There is a HUGE variability in viral replicative strategies. They pretty much can do anything we can imagine (as long as it does not break any physical laws:) ).
Yes, there is a special name: It's called persistent infection. It's a pretty general term, which may mean a lot of different things, but it assumes that infected cell is not killed immediately. I'll give you two examples:
Oncoviruses is a general name for viruses that make their host cell cancerous. They turn off the essential check-points in cell cycle and force cells to divide indefinitely. The cells release new viruses, not in huge amounts, but not "one-virus at a time" either. Certainly enough to infect new cells and new hosts.
HIV causes AIDS and is not an oncovirus. Normally it stops cell division and kills the cell within several days. But some strains of this virus don't kill their host cells. Cells feel sick for a couple of days and then recover and start dividing. They shed large amounts of virus all the time, but they continue to grow and divide. In vivo such cells get killed by immune system, but in culture you can keep these cells growing indefinitely.