I suppose it depends on which you consider more fundamental to the definition. The reason this particular ring and set of operations is interesting is because it implies being able to compute any Turing-computable function of the data. Hence Gentry's high-level overview paper was entitled "Computing arbitrary functions of encrypted data" [1].
[1] http://ece.gmu.edu/coursewebpages/ECE/ECE646/F10/project/F10...