Cannot delete an Inactive Apple ID from my MacBook Air

I have logged in my System settings using a Appleid which is inactive. Neither I can log out nor delete the appleid. I have a valid and active appleid which I can use here. But as I cannot log out from the existing one, I am unable to log in using another account. Most annoying is the system notification are coming again again promting solving the Applie problem. How can I get rid off this situation?

[Edited by Moderator]