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We study conditions on automorphisms of Boolean algebras of the form $\left(\lambda \right)/{ℐ}_{\kappa }$ (where λ is an uncountable cardinal and ${ℐ}_{\kappa }$ 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 $\left(2^{\kappa }\right)/{ℐ}_{\kappa ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M{A}_{\aleph ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial.