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(Λ∕M,Λ∕M) = 0. The technique of localization
now implies that gl.dimΛ = 0.
