On the Dimension of Modules and Algebras, IV: Dimension of Residue Rings of Hereditary Rings
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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 thenThe 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.
2018 ◽
Vol 9
(21)
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pp. 459
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2017 ◽
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2021 ◽
pp. 247263032110233
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1988 ◽
Vol 37
(3)
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pp. 353-366
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