Example where supp(M) is not Zariski-closed

Question

I found on a book that there exists R-modules M such that supp(M) is not Zariski-closed. I already know that if M is finitely generated then it is Zariski closed but I can't find an example of a non finitely generated R-module that is not Zariski closed. I found an answer on a book but I can't understand it well, it is too complicated and I'm quite new in this topic. I'm just looking for a simple example of this. Thank you