Short and simple question. Am I allowed to not include my email at a publication?
On a related note, what should I do if I don't have an e-mail address, or have an e-mail address at my university which will expire when I leave the university?
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You should put your email on your paper. Not doing so means people who want to contact you can't, and it's obnoxious, non-collegial, and extremely unusual in science.
As for what address to use, I see two main paths: "permanent" and "traceable":
For the "permanent" path: many universities have a permanent alumni email address, which you can set up while still there and which is an excellent choice for long-term usage. If you don't have access to such, you can get a free address (Gmail is a good "professional" choice, but others can be as well; don't use AOL, Hotmail, or others with a poor reputation). The advantage of this approach is that your address need never change; the disadvantage is that you may not be using an email associated with your primary affiliation.
For the "traceable" path: it is worth recognizing that even when you have a "permanent" position, such as a tenured faculty post, that it may not be permanent. People relocate away from long-term jobs for all sorts of reasons, and an email address thought to be forever is not. Everybody who wants to contact you will understand this. Thus, rather than use Gmail, you might instead just make sure you always maintain an easily identifiable web presence that includes information about your former affiliations and papers. Thus, when somebody searches for you, they can easily link you and your paper, even if your email has changed.
Whether you can choose not to include your email address in your paper depends on the journal. Some journals make it mandatory for the authors to include their e-mail addresses, while others do not mention anything about email addresses in their guidelines. Read the journal guidelines carefully; you can choose to not include your email address if the journal does not have any problem with it.
Having said that, I would say that you should provide some other detail that will allow readers to contact you. This is required for two reasons: first, to establish your credibility, and second, to be available to readers if they wish to contact you for some clarification. I can understand your concern that the university email is not permanent. However, you can definitely use a personal email address (although an institutional email looks more professional), or provide the link to your website or any other webpage that will provide your updated contact information.
If you are the only author, providing an e-mail address (or at least another form of contacting you) is highly adviseable, as explained by several of the other answers.
If you are not the only author, and some of the other authors can provide a more permanent e-mail address, then it depends on the conference or journal whether you can skip providing your e-mail address. If the conference or journal allows so, you do not need to provide an e-mail address, as the other authors already provide some contact information that is probably more reliable/durable than your own address. Otherwise, in the worst case, future readers might, of all of the indicated addresses, pick yours, with either of the following results:
The main reason I don't like putting my email address on the paper is because my inbox becomes rapidly flooded with requests to become a reviewer for some vanity press or international journal or an endless list of requests to publish in journals that are not PubMed indexed or charge money to submit.
What's more, I would prefer not to put my institution on their either, particularly if the work was not supported by my institution or reflects controversial views not sponsored by my employer. I haven't found a way around that, though.