Sharing Ideas without Intent to Coauthor
Most promotions in academia are based up letters of recommendation written by people who are not your coauthors. These people base their letters both on your published research and their general impression of you as a research mathematician. This general impression can be greatly improved by giving away ideas.
When attending a conference or a seminar, you may have an insight which would be worth sharing with relevant people just as a means of demonstrating that you are knowledgeable and find their work interesting. The insights you should consider giving away, without any expectation of coauthoring, are the ones which you would not have the time to investigate yourself anyway. You might, for example, notice that a presenter's work is related to a presentation you've seen in the past, but realize that linking these papers yourself would require reading two very difficult works or learning a new subfield. If you decide to give away an idea, send an email perhaps providing links to an exact paper. Be sure to use phrases like "I thought you might be interested..." to indicate that you are giving them an idea with no expectation of coauthorship. Spending an hour or two thinking over the mathematics before sending the email will maximize the chances that you are in fact making a useful suggestion that will lead somewhere. When the idea leads to a paper or an improvement of another person's paper, that is when you are most likely to be able to ask that person for a letter of recommendation in the future.
Another opportunity to add to other people's work is when a preprint is initially posted on the arxiv. If you subscribe to your field and can read papers relatively quickly, you will have the chance to provide a suggested improvement: perhaps even removing an hypothesis from one of the theorems. This is the ideal way to contribute for a well read mathematician who can keep track of many results. For the mathematician who is an expert at verifying proofs or concocting counter examples, this is an opportunity to catch an error in a paper and send an email suggesting that perhaps a certain hypothesis might have been overlooked. Error corrections are not received as well as suggested improvements, so be sure to have someone read over your email for the politest possible phrasing. The third possible contribution to an existing preprint is a suggestion of a future project: "this is an intriguing paper, do you think that, combined with the work of xy it might lead to results in the direction of z?" In fact you might cc xy when sending this email and take on the role of a mathematical matchmaker. This last kind of email is most effective when contacting people who might not have heard of each other's work. Remember, a short suggestion in an email like this will not be taken as a request to coauthor on such a project. Here we are talking about giving away an idea to improve your reputation in the mathematics community.
When giving away ideas, it is essential to be careful that you are giving away your own ideas. For example, you may have read a colleague's research proposal and then attended a talk by a speaker he invited to speak. Mentioning to the speaker that your colleague's work might be combined with hers to produce xyz is a major breach of confidentiality. Most likely, your colleague is well aware of the relationship and invited the speaker to learn more about her work without necessarily wanting the speaker to know how he planned to apply it. If the colleague wants the speaker to know, the colleague will tell her himself.
Finally, never give away ideas that you have already been discussing with potential coauthors. You may have decided you no longer have time for a project, but that coauthor may still wish to work on it. You should convey clearly to the coauthor that they can continue the project on their own and possibly suggest alternate collaborators. However, do not assume that just because you collaborate well with X and you collaborate well with Y, that ideas you discuss with X can be discussed with Y. They may not wish to work together, especially if you abandon the project and are no longer available as a unifying force.
