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We characterize the groups which do not have non-trivial perfect
sections and such that any strictly descending chain of non-“nilpotent-by-finite” subgroups is finite.
We characterize the solvable groups without infinite properly ascending chains of non-BFC subgroups and prove that a non-BFC group with a descending chain whose factors are finite or abelian is a Cernikov group or has an infinite properly descending chain of non-BFC subgroups.
We characterize left Noetherian rings which have only trivial derivations.
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