"[DB92A95EE8E5]" . . . "http://dx.doi.org/10.1515/FORM.2011.101" . "\u0160aroch, Jan" . . "11320" . "DE - Spolkov\u00E1 republika N\u011Bmecko" . "Kaplansky classes, finite character and aleph(1)-projectivity" . . . . . "Trlifaj, Jan" . "Kaplansky classes, finite character and aleph(1)-projectivity"@en . "24" . . "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 C of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair (C, {=) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = C-perpendicular to' for a class C' 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 (C, {=) is an AEC of roots of Ext, then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph(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 (perpendicular to)(D-perpendicular to) equals the class of all flat modules and consists of modules of projective dimension {= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non-right perfect ring R." . . "RIV/00216208:11320/12:10128138!RIV13-GA0-11320___" . . . . . "2"^^ . . . "19"^^ . "144348" . "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 C of roots of Ext is a Kaplansky class closed under direct limits if and only if the pair (C, {=) is an abstract elementary class (AEC) in the sense of Shelah. We prove that this AEC has finite character in case C = C-perpendicular to' for a class C' 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 (C, {=) is an AEC of roots of Ext, then C is known to be a covering class. However, Kaplansky classes need not even be precovering in general: We prove that the class D of all aleph(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 (perpendicular to)(D-perpendicular to) equals the class of all flat modules and consists of modules of projective dimension {= 1. Assuming the Singular Cardinal Hypothesis, we prove that D is not precovering for each countable non-right perfect ring R."@en . "P(GA201/09/0816), P(GD201/09/H012), P(MEB101005), Z(MSM0021620839)" . "000309161800008" . . "10.1515/FORM.2011.101" . . "2"^^ . "Forum Mathematicum" . "0933-7741" . "RIV/00216208:11320/12:10128138" . . "Kaplansky classes, finite character and aleph(1)-projectivity" . . "5" . . . . "Kaplansky classes, finite character and aleph(1)-projectivity"@en . "aleph1-projectivity; character; finite; classes; Kaplansky"@en . .