Is it possible to be a mathematician without being mathematically talented? Apparently I'm not mathematically talented enough to be hired as a postdoc. Is there any way to earn a living while continuing to do mathematics I love (commutative algebra)? I don't like teaching or working in financial market.
 A: There was a theoretical physicist you may have heard of who once earned a living working in a patent office.
A: Of course you can be a mathematician.  Robert Ammann, Marjorie Rice, and Pál Turán all did spectacular mathematics while a postal clerk, housewife, and Nazi prisoner of war.
Pál Turán was being worked to death under the harshest of conditions in Nazi germany. He worked out results in extremal graph theory in his mind while enduring forced labor. Whenever he could find a scrap of paper, he wrote down some of his results. 
Being a mathematician is a state of mind.
A: If you want to get paid to do research on topics that you choose, then academia is your only choice.
If you are finishing a PhD and want a career in academia, you have two real choices : get a postdoc, or find a position at a teaching-oriented institution.  If your track record is not strong enough to get a postdoc and you don't enjoy teaching, then neither of these are options.  There are other things that people do which allow them to hang on for a short time (eg 1-year visiting positions or adjuncting), but these are exploitative and I strongly advise you not to waste your time on them.  All you will end up doing is wasting a couple of years before entering the non-academic workforce.
There are really no jobs outside of academia which will allow you to continue doing your research.  However, there are jobs in which you can continue doing math in some form or another.  For instance, if you are a US citizen, then you can work for the NSA.  They will tell you what to work on, but you will still be doing math.  You could also work for a high-tech firm or in finance.  You won't be proving theorems, but you will still be solving problems and using your mathematical skills.
A: Well, there are many universities, and they're not equally competitive. Getting a postdoc position at, say, Michigan State will be easier than getting one at MIT, although it will still be very difficult!
Outside of academia it's rare to be paid for doing research in pure mathematics, particularly in abstract algebra. It seems to have fewer applications than most other fields - fewer even than number theory (applied to cryptography), combinatorics and graph theory (applied to algorithm design) or category theory (applied to programming language design).
One option would be to look at companies which do paid research in mathematics - one example is Think Tank Maths who are based in Scotland, but there must be similar companies in the US.
A: Yes, you can earn a living while doing commutative algebra. If you can get an undemanding job you may have a lot of free time to do math; this happened to me once. (You wouldn't be getting paid to do commutative algebra of course.) In some countries it's easier to pull this off this than others. Government jobs, and to a lesser extent government contractor jobs, have the reputation of having a lot of "downtime", but of course this varies widely from job to job. Maybe some other posters would know of other easy jobs where this can be done.
