  We establish rigidity for partial transformation groupoids associated with
algebraic actions of semigroups: If two such groupoids (satisfying appropriate
conditions) are isomorphic, then the globalizations of the initial algebraic
actions rationally embed in each other. For specific example classes arising
for instance from toral endomorphisms, actions from rings, or actions from
commutative algebra, this mutual embedability can be improved in various ways
to obtain surprisingly strong rigidity phenomena. This is witnessed in a
particularly striking fashion for actions arising from algebraic number theory:
We prove that the groupoids associated with the action of the multiplicative
monoid of non-zero elements in a ring of algebraic integers on the additive
group of the ring remembers the initial algebraic action up to isomorphism,
which in turn remembers the isomorphism class of the ring. This resolves an
open problem about isomorphisms of Cartan pairs and leads to a dynamical
analogue of the Neukirch--Uchida theorem using topological full groups.
