What additional privileges does the root user have over a standard user in Linux?
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The root user can do pretty much anything that the hardware allows. The root user can write directly to hard drives without going through the filesystem. Root can modify the kernel via modules. The root user can also circumvent any security policy of the system software: he can change ownership of any files, set up privileged ports, write to privileged filesystem locations, gain extra scheduling priority levels, etc. The root user can also spy on any other user in the system.
Regular users can generally only manipulate their own processes and their own filesystem subtrees. They tend to be forbidden from manipulating the system's hardware, security policies, and from meddling with other users' files and processes.
On Unix, users are represented by a number (User ID, or UID). Any user with UID 0 is privileged, normally there is only one called
Unix system permissions are more or less represented as permissions (the typical rwx) on files (devices, like disks, are also files for Unix). One of the differences is that root can override any read and write permission.
Another permission area is the manipulating of processes, something usually only the owner of the process (typically whoever started it, unless it is a SUID executable) is allowed to signal a process or attach to it for debugging. Again, root can override this.
Some operations (like powering off or rebooting) explicitly check for root.
Note too that modern security frameworks can be used to limit and grant privileges in a separate, fine-grained form. See this thread for some details on machines with
root with no password, open to the whole 'net.
root is the system administrator, and they can do whatever they want to the system, which can lead to data corruption if the account is not used wisely. Since it appears you are new to Linux, I would recommend you not use root until you get more experience with Linux. To access an administrator/root command prompt from a standard users command prompt, type
su (I believe most distros support this) and then hit enter.
Beware when using root, as you can really mess up your computer.