Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal? I am a neophyte amateur mathematician. I have been reading a lot about journals and the topic of peer-review in mathematics journals. Does one have to have professional credentials or have a Doctorate in order to publish in peer-reviewed mathematics journals or just the desire to compently solve mathematical problems?  

There is usually no technical requirement for any particular credentials, and it seems unlikely to me that editors and referees will check for this. The usual publication process is that you submit a paper to a journal editor, who then forwards it one or more competent referees, who make a recommendations on the suitability for publication. Often, the referees make suggestions for improving the paper, whether or not it is found acceptable for publication.
Nevertheless, an amateur mathematician will find him or herself at a disadvantage in several respects of this process. First, for someone with less experience in the professional mathematical community, it may be more difficult to recognize which accomplishments actually merit publication, and editors sometimes receive articles submitted in good faith from amateurs, which are trivial or seriously wrong in some respect. Second, even when the result is correct and worthwhile, if the presentation of the paper deviates from the accepted norms, it may be found wanting. For example, it is more difficult for an amateur to know which topics need more careful explaining in a paper and which do not, and inappropriate decisions in such cases can hurt the reception of the paper. 
I see (as I write this) that others have now given some concrete advice. The most important advice I can give is to take the suggestions of the editors and referees seriously. If someone claimes that part of your paper is unclear or wrong, then you should try to understand exactly why they thought so; doing so will inevitably lead you to a better paper.
