That is a most unfortunate, and terribly common, complaint. It is known as tinnitus, and affects a very high number of people, especially those with hearing loss due to other causes, such as ear infections or hearing loss due to listening to loud sounds for too long. The problem stems from the brain being used to hearing a certain baseline level of noise, and then interpreting that to be recognizable sound. When there is no sound at all, the brain does not know how to respond, and interprets the silence as ringing or buzzing noise that is incessant. That is why the ringing actually seems to get worse when you are in a quiet room, or when your hearing aids are out: the sound isn't real, it is your brain interpreting the absence of sound.
There are several things that can be done about this condition, but there are no perfect solutions at this point. Depending on how bad the problem is, an excellent place to start would be with visiting an ear-nose-and throat surgeon
) who can perform some hearing tests and then make recommendations. Some options include medications, and even the addition of "white noise" via special aids made for this purpose. There is also much research going into this area now, and hopefully there will be better treatments in the future.