Presenting and disseminating research results before submitting a paper I got a PhD in math in another country years ago. Recently I moved to the US and have a tenured position. Assume I wrote a paper and want to publish it in a journal. Is it considered OK to present the work and/or post an electronic copy on arXiv, before submitting the paper to a journal?  In particular, does it matter in what order I do the following things?

1) Present the work in a seminar at my department.

2) Present the work in a conference in my field.

3) Present the work in a seminar at another department.

4) Post the paper on the electronic arXiv.

5) I submit the paper to a journal.
Overall, I don't think there are much such rules; but I am not in the US, so my expertise is limited.

I personally refrain from presenting a work in seminars (in other department notably) or conferences before it is on the arXiv. The deposit is indeed the moment I have checked everything, and I am confident it holds together. Before that, I may have doubts.

In rare occasion, the preprint takes some time to be finished, or deposited, and I am still confident enough to present it, but it is very rare.

I usually put it on the arXiv before submission, because it is a time stamp and I want my work to be available. Referees rarely check proofs in detail, so you need to make yourself confident about your work anyway. You should however remember to update the preprint to a so-called postprint when the referee reports come in (note that some publishers inexplicably don't allow that, OUP for example).

Another reason to deposit on the arXiv before submission is that Elsevier allows you to update a preprint to a postprint, but is quite unclear whether you have the right to deposit a postprint or preprint for the first time after acceptance. Well, not that I particularly urge anyone to publish with Elsevier, but that is to be known.