caa a22 4500 388023783 CHVBK 20180307124931.0 cr unu---uuuuu 161130e199802 xx s 000 0 eng 10.1017/S0013091500019404 doi S0013091500019404 pii (NATIONALLICENCE)cambridge-10.1017/S0013091500019404 The Cartan determinant and generalizations of quasihereditary rings [Elektronische Daten] The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the "projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them. Copyright © Edinburgh Mathematical Society 1998 Burgess W. D. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada, K1N 6N5, E-mail address: wdbsg@uottawa.ca Fuller K. R. Department of Mathematics University of Iowa Iowa City, IA, USA, 52242, E-mail address: kfuller@math.uiowa.edu Proceedings of the Edinburgh Mathematical Society Cambridge University Press 41/1(1998-02), 23-32 0013-0915 41:1<23 1998 41 PEM https://doi.org/10.1017/S0013091500019404 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0013091500019404 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Burgess W. D. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada, K1N 6N5, E-mail address: wdbsg@uottawa.ca NATIONALLICENCE 700 1- Fuller K. R. Department of Mathematics University of Iowa Iowa City, IA, USA, 52242, E-mail address: kfuller@math.uiowa.edu NATIONALLICENCE 773 0- Proceedings of the Edinburgh Mathematical Society Cambridge University Press 41/1(1998-02), 23-32 0013-0915 41:1<23 1998 41 PEM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge