In my number theory class, we're learning about Quadratic Residues (QRs) vs. Nonquadratic Residues (NRs, also sometimes called quadratic nonresidues) in $\mathbb F_p$. It works out that multiplying two QRs together gives another QR, as does multiplying two NRs. Multiplying an NR and a QR gives back a NR. My number theory prof encouraged us to think about it like multiplying positive and negative numbers, since similar rules apply.
This has gotten me thinking about applying similar concepts to other rings. Given any ring $R$, we can say that a nonzero element $r \in R$ is a QR if there exists some $s \in R$ such that $s^2 = r$, and that it is an NR otherwise. I've been thinking a bit about it, but I haven't been able to find any nice general patterns, nor could I find anything about it online.
In $\mathbb R$, the QRs are the positive numbers and the NRs are the negative numbers, and the multiplication rules described above still hold here. However, in other fields, like $\mathbb Q$, the multiplication rules don't work the same (i.e. $2 \times 3 = 6$, but none of those are squares in $\mathbb Q$). This also doesn't hold in $\mathbb Z$ for the same reason.
There are also some rings where we seem to only have QRs. Aside from the trivial ring, any algebraically closed field will only produce QRs (solve $x^2-c=0$ to show $c$ is a QR). Also, in $\mathbb F_4$, all elements of that field are the square of another element, even though that field is not algebraically closed.
Is there any generalized notion of this concept to other rings? I've tried searching online and I haven't been able to find anything.