A Ring of Dimension Door wouldn't work(RAW), but a (Greater, Minor) Ring of Spell Storing with Dimension Door(s) in it would.
Activating a magic item is not 'casting' per this (emphasis added) line from Dimensional Dervish
You can take a full-attack action, activating abundant step or casting dimension door as a swift action.
If you want to have more uses per day than you have spell slots/Ki for Abundant Step, you could use a Ring of Spell Storing which indicates
A ring of spell storing contains up to 5 levels of spells (either divine or arcane, or even a mix of both spell types) that the wearer can cast...
That all said, it's worth petitioning the GM for; there is little chance of it breaking a game and could go a long way toward fulfilling certain character concept goals.
Of course neither item qualifies you for the feat(s), but that doesn't seem to be what your question is referring to.