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On the Dimension of Modules and Algebras, IV: Dimension of Residue Rings of Hereditary Rings

Published online by Cambridge University Press:  22 January 2016

Samuel Eilenberg ,
Hirosi Nagao  and
Tadasi Nakayama
Samuel Eilenberg
Affiliation:
Columbia University
Hirosi Nagao
Affiliation:
Osaka City University
Tadasi Nakayama
Affiliation:
Nagoya University
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A ring (with unit element) Λ is called semi-primary if it contains a nilpotent two-sided ideal N such that the residue ring Γ = Λ/N is semi-simple (i.e. l.gl.dim Γ = r.gl.dim Γ = 0). N is then the (Jacobson) radical of Λ. Auslander [1] has shown that if Λ is semi-primary then

The common value is denoted by gl. dim Λ. On the other hand, for any ring Λ the following conditions are equivalent : (a) 1. gl. dim Λ ≦ 1, (b) each left ideal in Λ is projective, (c) every submodule of a projective left Λ-module is projective. Rings satisfying conditions (a)-(c) are called hereditary. For integral domains the notions of “hereditary ring” and “Dedekind ring” coincide.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 10 , June 1956 , pp. 87 - 95
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1956

References

[ 1 ] Auslander, M., On the dimension of modules and algebras, III, Nagoya Math. J. 9 (1955), 67–77.CrossRefGoogle Scholar
[ 2 ] Cartan, H. Eilenberg, -S., Homological Algebra, Princeton Univ. Press, 1956.Google Scholar
[ 3 ] Eilenberg, S., Algebras of cohomologically finite dimension, Comment. Math. Helv., (1954), 310–319.Google Scholar
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