This is not Adobe. Rather user to user. All of us here are visitors who visitor according to our personal schedules. No instant responses, although I try to see that everyone has a viable answer to his/her question as quickly as possible.
The answers are in the details. And, not much to go on in what you wrote. So, here comes the questions.
1. What version of Premiere Elements do you have? Is there an error message associated with the failure to install with your serial number?
Are you doing this install with antivirus and firewall(s) disabled?
2. What is the operating system of the old computer and what is the operating system of the new computer?
3. Did you purchase Premiere Elements and its serial number as online downloaded installation files from Adobe or from installation disc
that was in box packaging of the product?
4. When you started out, did you have a redemption code or the actual serial number?
Let us start here and then decide what next.