Variations on a theme of Kaplansky

Kaplansky's lemma says, for bounded linear operators on Banach spaces, that locally algebraic implies algebraic. The proof divides into two orthogonal components, one of which uses Baire's theorem and the other the Euclidean algorithm. We examine both these arguments, show the failure of a possible dual to Kaplansky's lemma, and discuss the extension of the discussion to whether or not an operator has finite ascent.