Let \(B = \bigoplus _{i \in G} B_i\) be a commutative integral domain of characteristic 0 graded by an abelian group G. We say that B is rigid (resp. graded-rigid) if the only locally nilpotent derivation (resp. homogeneous locally nilpotent derivation) of B is the zero derivation. Given a subgroup H of G, define \(B^{(H)} = \bigoplus _{i \in H} B_i\) . We give results that answer or partially answer the following questions: Does non-rigidity of B imply non-rigidity of \(B^{(H)}\) ? When can a derivation of \(B^{(H)}\) be extended to one of B? What are the properties of the set of subgroups H of G such that \(B^{(H)}\) is not graded-rigid? We define the subgroups \({\bar{{\mathbb {G}}}}(B) \subseteq \mathbb {G}(B)\) of G and find that these are related to the locally nilpotent derivations of B in several interesting ways. (The definitions of \({\bar{{\mathbb {G}}}}(B)\) and \(\mathbb {G}(B)\) do not involve derivations, and these two groups are usually easy to determine.) One of our results states that if B is a normal affine G-graded domain then \({\text {trdeg}}(B: {\text {ML}}(B)) \geqslant {\text {rank}}( \mathbb {G}(B)/{\bar{{\mathbb {G}}}}(B) )\) . We also give a result relating the rigidity of \(B_{(x)}\) to that of B/xB, where B is an \({\mathbb {N}}\) -graded normal affine domain and x is a homogeneous prime element of B. We give some applications to Pham-Brieskorn rings.
