It looks like you installed the root user using the rvm installation, which is located in your home folder. This means that when you install gem with sudo, the files will be owned by root. So, when you try to install a gem like you, you get permission errors.
To fix this, you need to return the rvm folder to you (I assume that from your file paths your username is "da" here) ...
sudo chown -R da:da /Users/da/.rvm
To avoid this in the future, I would recommend installing the root path of both your own rvm installation and system planes.
idlefingers
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