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The Cartan determinant and generalizations of quasihereditary rings

Published online by Cambridge University Press:  20 January 2009

W. D. Burgess  and
K. R. Fuller
W. D. Burgess
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada, K1N 6N5, E-mail address: wdbsg@uottawa.ca
K. R. Fuller
Affiliation:
Department of MathematicsUniversity of IowaIowa City, IA, USA, 52242, E-mail address: kfuller@math.uiowa.edu
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Abstract

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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 1 , February 1998 , pp. 23 - 32
Copyright
Copyright © Edinburgh Mathematical Society 1998

References

REFERENCES

1.Agoston, I., Dlab, V. and Wakamatsu, T., Neat algebras, Comm. Algebra 19 (1991), 433–442.Google Scholar
2.Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, second edition (Springer-Verlag, New York, Heidelberg, Berlin, 1992).Google Scholar
3.Burgess, W. D. and Fuller, K. F., On quasihereditary rings, Proc. Amer. Math. Soc. 106 (1989), 321–328.Google Scholar
4.Burgess, W. D., Fuller, K. R., Voss, E. and Zimmermann-Huisgen, B., The Cartan matrix as an indicator of finite global dimension for artinian rings, Proc. Amer. Math. Soc. 95 (1985), 157–165.Google Scholar
5.Dlab, V. and Ringel, C. M., Quasi-hereditary algebras, Illinois J. Math. 33 (1989), 280–291.Google Scholar
6.Fuller, K. R., The Cartan determinant and global dimension of artinian rings, Contemp. Math. 124 (1992), 51–72.Google Scholar
7.Fuller, K. R., Algebras from diagrams, J. Pure Appl. Algebra 48 (1987), 23–37.CrossRefGoogle Scholar
8.Gustafson, W., Global dimension in serial rings, J. Algebra 97 (1985), 14–16.Google Scholar
9.Hoshino, M. and Yukimoto, Y., A generalization of heredity ideals, Tsukuba J. Math. 14 (1990), 423–433.CrossRefGoogle Scholar
10.Uematsu, M. and Yamagata, K., On serial quasi-hereditary rings, Hokkaido Math. J. 19 (1990), 165–174.CrossRefGoogle Scholar
11.Yamagata, K., A reduction formula for the Cartan determinant problem for algebras, Arch. Math. 61 (1993), 27–34.Google Scholar
12.Zacharia, D., On the Cartan matrix of an artin algebra of global dimension two, J. Algebra 82 (1983), 353–357.Google Scholar