Vol. 26, No. 3, 1968

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Simple modules and hereditary rings

Abraham Zaks

Vol. 26 (1968), No. 3, 627–630

The purpose of this note is to prove that if in a semi-primary ring Λ, every simple module that is not a projective Λ-module is an injective Λ-module, then Λ is a semi-primary hereditary ring with radical of square zero. In particular, if Λ is a commutative ring, then Λ is a finite direct sum of fields. If Λ is a commutative Noetherian ring then if every simple module that is not a projective module, is an injective module, then for every maximal ideal M in Λ we obtain Ext1∕M,Λ∕M) = 0. The technique of localization now implies that gl.dimΛ = 0.

Mathematical Subject Classification
Primary: 16.40
Secondary: 18.00
Received: 5 December 1967
Published: 1 September 1968
Abraham Zaks