It's not in your PRIA record because there is no such thing. PRIA is a system which requires airlines to obtain records from the FAA and previous employers.

You would be required to report a previous 121 employer when you fill out the PRIA forms, and a new employer would then obtain documentation from PSA indicating that you were fired for ______.

The cause of the termination will be reported...presumably that will be poor training performance. The fact that no checkrides were failed simply means your performance in training was so bad that the instructor did not want to sign an 8710. Or maybe the instructor hated your guts (it happens).

Bottom line, yes you have to report it, yes the new employer will get the records, and yes this will make it hard for you to get a 121 job. The best way to overcome this is go fly some 135, hopefully some turbine/crew operations and build more time. After you build a history of passing 135 rides, the regionals will look at you again.