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Kaplansky classes emerged in the context of Enochs' solution of the Flat Cover Conjecture. Their connection to abstract model theory goes back to Baldwin et al.: a class    of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair    is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case    for a class    of pure-injective modules. In particular, all AECs of roots of Ext over any right noetherian right hereditary ring  R  have finite character (but the case of general rings remains open). If    is an AEC of roots of Ext, then    is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class    of all ℵ 1 -projective modules (which is equal to the class of all flat Mittag-Leffler modules) is a Kaplansky class for any ring  R , but it fails to be precovering in case  R  is not right perfect, the class    equals the class of all flat modules and consists of modules of projective dimension   . Assuming the Singular Cardinal Hypothesis, we prove that    is not precovering for each countable non-right perfect ring  R .

Kaplansky classes, finite character and ℵ 1 -projectivity