I have an idea which I'd like to work on with a professor at my university. In my initial email would it be arrogant to ask if he's interested in working with me on this particular topic (he's published some papers on closely related topics), or should I just say that I'm interested in generally doing research with him? I really do want to work on this particular idea but also don't want to come across as trying to tell him what to do.
A small point about language: in the U.S., in mathematics, it might be more tactful and less presumptious (also) to not ask about "work with me", but "advise me on" [the project you propose]. At least to my perception (=old guy) this leaves many more options for the precise relationship. E.g., experienced people can often and easily give helpful advice without really actively participating. At the same time, depending..., they might find interest in doing so. Or, at the same time, they might see that it's better to let you do it yourself... etc.
I think that most professors like to hear about ideas that their students have to work with them.
I would simply send him an email telling him that you have an idea that you want to work on, and that you think he might be interested in.
Probably best if you talk about it face to face as most professors are too busy to read long emails.
Some thing like this:
Dear Prof X
I am contacting you as I recently had an idea for a research project on topic x.
I really enjoyed working with/learning from you on x [something he did, e.g., a class], and I am very interested in x [something he researches].
With that considered I was wondering if you would be interested in working with me on this project?
If you would like to know more, I would love to have a meeting at a time that is convenient for you.