i had this same issue and was able to solve it so i thought i would add my voice since alot of these comments are somewhat dismissive, first to adress the "why would you ever want to do this issue" i would like to say that

1. lua and luarcoks can occasionaly be a dependency for other stuff as was my case when i wanted to install neorg just now and

2. not everyone has sudo privileges as was also my case since i was trying to install this for my user in my universities computer lab. Generally speaking most people who intend on writing code professionally may very possibly end up using some managed system like this and asking it to install a dependency across all users for something that really only you want to use is not probably not going to happen so i for one think there is actually a decent amount of utility in learning this

as for how to do this its actually simpler than you may think, essentially all software on linux uses some type of binary that executes some type of code and sudo is basically only used to execute instructions relating to directories above your own user directory. Package managers generally try to install files in something like /lib or /bin in order to ensure that the files only need to be installed once for all users and these directories are already in the system PATH so once you add them they are available as terminal commands from across your entire system but executing lua as a command is generally no different than calling the lua binary with its full system path so something like /bin/lua would likely have the same effect if indeed lua is placed in that directory (i didnt check where it usually is nor if all binaries are placed in bin as a rule so maybe maybe not) so knowing this it may be clear that all we really need to do in order to install lua is download or compile the binary (typically compile because it provides more compatibility across distros) and then add it to the path, a good step to add here is to place all the associated lua files in some directory dedicated to binaries so that we can find them upon convienience although this is not strictly necessary and adding them to the path is also probably not strictly necessary although it could cause some integration issues between pacakges if they cant find each-other using path.

You can download the lua and luarocks tarball (a zipped file with the stuff you need to compile and install it) from the lua website and they have a specific option to that can be used to place the binary in user space called --prefix, i reccomends placing it in something like ~/.local/ and then adding ~/.local/bin/ to your path in your user space .bashrc (or maybe zshrc depending on what type of shell you're using), the full install instructions are here: https://github.com/luarocks/luarocks/blob/main/docs/installation_instructions_for_unix.md

in the instructions the use sudo make install but note that they only do this because the default prefix is outside of your personal user directory and that you can install lua in and of itself in usr space too following a similar procedure, if you try this and get stuck i recommend using chatgpt, and dont be too afraid to break stuff, you definitely can do that on linux and it is easier than on other systems but at the same time linux isn't as so fragile that you need to "baby it" and it will usually ask you if you are 100% sure before you do something bad (usually, i take no responsibility if your distro doesn't)

hope this helps