Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$.  

A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4).
(<http://en.wikipedia.org/wiki/Algebra_over_a_field#Classification_of_low-dimensional_algebras>). Now where is F? Not there in this form, because by a basis change it's isomorph to $A\bigotimes{A}=E$ (and that is listed). But the two are different fusion rings! 


Thus: Are the "simple objects" of a fusion ring "marked" somehow and basis change is strictly forbidden?



No basis change in a fusion ring allowed?
That's right, fusion rings have more data than just being a ring, they also have a preferred basis corresponding to the simple objects. So two fusion rings can be isomorphic as rings but not as fusion rings.


