The theory of ring C*-algebras was initiated by Cuntz and developed by Cuntz?Li and Li. The construction of reduced ring C*-algebras is subsumed by a general C*-algebraic construction from algebraic actions: Each algebraic action of a semigroup gives rise to a concrete C*-algebra generated by the left regular C*-algebra of the group being acted on together with the Koopman representation for (the dual of) the semigroup action. Reduced ring C*-algebras arise from this construction by taking the monoid of left regular elements of a ring and letting it act on (the additive group of) the ring by left multiplication. I will present on joint work with Xin Li in which we find groupoid models for the concrete C*-algebras arising from non-automorphic algebraic actions. We then use these models to obtain structural results for the C*-algebras (even for the case of ring C*-algebras, we improve on all previous results). Surprisingly, the concrete C*-algebras in question may be exotic groupoid C*-algebras. The groupoids from our construction turn out to exhibit a rather surprising form of rigidity, which I will briefly discuss.