If someone was arrested in the state of Rhode Island but never charged with a crime, will this show up in a national background check used by employment agencies for hiring??

Arrests generally do not show up, only convictions.

Normally not, but in today's computer world all things are possible, remember if there was a notice of the arreest in a news paper that was online, it is out there forever to be found.

But normal checks only show convictions not arrests unless it is done for law enforcement or national security background checks.