Can undergraduates publish notes (or articles) in journals such as Mathematics Magazine, or does one need a PhD? Will the paper be rejected? I could not find any information about this online.
Edit: How long does the referring process take? When will I heard from the journal?
On
I think Involve specifically welcomes articles by undergraduates.
Here is a link to a list of such journals provided by Alex J Best.
Of course, you can submit your work to any journal. I am sure there are examples of undergraduate students who published in Annals of Maths. Well, actually, I'd like to see a concrete example.
But it takes very talented undergraduate students to publish in standard peer-reviewed journals publishing original research. It is more likely that undergraduate work finds its place in journals which are oriented towards unergraduate work.
Absolutely anyone can have work published. I published my first article as a PhD student, so I didn't have a PhD at the time.
Once submitted, the work will go to an editor who passes it on to a referee (or referees) that s/he thinks are suitable to review the article. This process is called "peer review". The referee (or referees) will decide if they feel the article is of sufficient quality and originality to be published. The referee will then pass his suggestion on to the editor who takes the final decision.
Of course, the required standard varies between journals.