I'm guessing that you have installed Lightroom 4.1 RC. This is a release candidate, but it's acting more like a beta in that no serial number is required. I suspect that when Lightroom 4.1 is officially released the serial number will be required.
Ha, I just asked the same question. I have installed the 4.1 RC as well and am not being promted to enter my newly acquired serial number. I guess I'll just go on editing. Hopefully the 4.1 final will come out soon (with all the fixes) and we can get on our way.
Your serial number is not erased by uninstalling Lightroom. The serial number from the Beta is stored in a file on your computer, and it may be valid for you full version of Lightroom, but it will expire and give you trouble later.
The best thing to do is go to Help > Lightroom Registration. Likely, you will see name and address fields. Ignore this and look at the left hand column:
Licese or Try
Go to License and you will have the boxes to enter your serial number. Simply enter your serial number and click "Finish".