How can one's career as a reviewer be ended?
No, the answer is simply that @buffy is wrong. In reality, editors work hard to find anyone willing to review a paper. They will be hesitant to exclude anyone. In most fields reviewers are anonymous, so only one publisher will know if a reviewer does a bad job.
The answer-fragment OP quoted is incorrect. Making a mistake while reviewing is not likely to lead to the end of your reviewing career. This comes from several angles (we neglect the possibility that the editors are hard-pressed to find reviewers):
In most fields you can't realistically be expected to verify everything in the paper yourself. For example if you receive a paper about a new discovery at the Large Hadron Collider, you can't be expected to build your own Large Hadron Collider, run the experiments yourself, and verify the discovery. It's simply not possible.
In most fields, there is some level of good faith assumed between the authors and the journal. The journal will not assume the author is actively attempting to deceive them (until proven otherwise). They will assume the author did perform the experiment. Therefore if you accept a paper that turns out to be a fraud, nobody is going to hold it against you.
Finally, only the journal that you review for is likely to know your identity. No other journal will know (unless you go public). It's possible editors will tell each other not to invite a certain reviewer, or perhaps if they are sharing the same reviewer pool, but there's no central repository of "bad reviewers" or anything like that.
In practice you'll only start receiving fewer reviewer invitations (i.e. reviewing career ended) if:
- You retire or pass away.
- You make it known that you're not reviewing anymore, e.g. with a notice on your website.
- You become research inactive, e.g. by not publishing new papers for a while.
Well, not feeling obliged to review other people's papers anymore sounds like a nice deal so if you figure out an answer tell me.
Unfortunately, there won't be one. Not only do I know plenty of mathematicians who have approved papers with errors in them but I've known a number of mathematicians who were well-respected in the community despite the fact that everyone figured there was something like a 1 in 8 chance that the main claim of any paper of theirs would turn out to be fatally wrong. I can't say for certain that I've done it since obviously I would have flagged the error if I'd seen it and errors are so common that no one even mails the author for non-critical errors and a reviewer likely won't even be told if a fatal flaw in the proof is later found.
Hell, I've been halfway through extending people's published work only to email the author a question and find out that the proof is in shambles and they are struggling to find a patch. So it's literally the exact opposite situation where the total absence of errors is what would be unusual.
Indeed, I don't know anyone who has reviewed more than one or two math papers who hasn't approved a paper with an error. Studies suggest that something like 80% of published math papers contain some form of error (that's not a fatal error but still). Sorry if I don't remember the source on that study but I'm sure if you google it you can find the relevant info.
Note that I think this is a compelling reason that mathematicians should completely abandon the blind peer review process in favor of something like a math social network with up and down votes. Yes, still have two independent individuals read the paper and submit comments and demands for clarification but don't throw out all that the reviewers have learned by collapsing the judgement down to accept/reject. The mathematician I was thinking of with the frequent errors still did good work but often pursued proofs that were particularly knotty and difficult to check. The reviewers were well aware that certain parts of these proofs raised yellow flags but they couldn't specifically show there were any flaws and, since tenured professors aren't always willing to break things down to a tedious level or formality, I agree publication was the right call. However, a math social network could have passed along the reviewer's sentiment that they still have some reservations about the argument in part X moreover, the initial review will matter less since the accumulation of comments and the ability to use all professional mathematical readers of the paper as a crowdsourced continuing review will do more to help us build a mathematical edifice we are sure is true.