EN
We study conditions on automorphisms of Boolean algebras of the form $𝓟(λ)/ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $𝓟(2^{κ})/ℐ_{κ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $MA_{ℵ₁}$ implies both that every automorphism of 𝓟(ℝ)/Fin is trivial on a cocountable set and that every automorphism of 𝓟(ℝ)/Ctble is trivial.