Undecided Question: Is school work covered by intellectual property law?

There are plenty of examples of intellectual products that are covered by Intellectual Property laws (which vary from country to country, and perhaps even between states). Examples of those can be: music compositions, software, writing, etc. This might also lead to tangent subject such as patentability, but that's beside the subject of this question.

On the other hand, I'm under the impression that in academia the researchers don't have exclusive rights over their work. Instead it belongs to the institute. In case of articles published in peer reviewed journals, the authors sign the articles but the property rights are transferred to the publisher; the same happens to books, whose rights are transferred to the publisher given a compensation to the authors. In other words, the authors let go of the rights to (re)publish their work through a different publisher without previous consent (which makes sense).

However, I'm not sure to WHOM belongs work that is produced under a school project (i.e. part of academic curriculum). If someone on the University produces a brilliant piece of math, or software, as result of their school project, does it belong to the school or to its creator? Are intellectual property rights tacitly attributed to the author, or does its rights transfer to the institute/university after the work is submitted (irrespective of the grade attributed by the course teacher)?

Instead it belongs to the institute.
I'm thinking, for example, of a researcher who creates a means of identifying the presence of a certain protein in blood samples and thus claims authorship (although the work "belongs" to the institute the researcher works/worked for).

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