Rigidity of Graded Integral Domains and of their Veronese Subrings

A ring R is said to be rigid if the only locally nilpotent derivation of R is the zero derivation. Let G be an abelian group, and B = (direct sum of B_i for i in G) be a G-graded commutative integral domain of characteristic 0. For each subgroup H of G, consider the Veronese subring B(H) of B, defined by B(H) = (direct sum of the B_i for i in H). We study the following questions. If B is non-rigid, does it follow that B(H) is non-rigid? Can derivations of B(H) be extended to derivations of B? What are the properties of the set of subgroups H of G such that B(H) is non-rigid?