Here's why:

* Au represents the element Gold.

* I represents the element Iodine.

* The subscript 3 indicates there are three iodine atoms per gold atom.

* (III) in the name indicates that the gold atom has a +3 oxidation state. This is because iodine has a -1 oxidation state, and to balance the three iodine atoms (-3 total charge), the gold atom must have a +3 oxidation state.

Therefore, the full and correct name for AuI₃ is Gold(III) iodide.