I was talking about a probable research topic that I got in mind for following a Phd in Computer Science to one of my past Lecturers. This professor told me that I should make a research proposal and see what happens with the evaluating committee. The research proposal was made and it got finally accepted. Now the issue is that this university, in which they accepted my proposal is not so well known worldwide. So I would like to apply to another more renewed university with the same topic, I think that could help me better in my academic career. What should I do? It is unethically to move to another university with my research idea? maybe my past professor could feel like betrayed?