Simple modules and hereditary rings

The purpose of this note is to prove that if in a semiprimary ring A 9 every simple module that is not a projective Λ-module is an injective Λ-module, then A is a semi-primary hereditary ring with radical of square zero.In particular, if A is a commutative ring, then A is a finite direct sum of fields.If A 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 A we obtain Ext 1 {AjM, AjM) = 0.The technique of localization now implies that gl.dimA = 0.1* We say that A is a semi-primary ring if its Jacobson radical N is a nilpotent ideal, and Γ = ΛjN is a semi-simple Artinian ring.Throughout this note all modules (ideals) are presumed to be left modules (ideals) unless otherwise stated.For any idempotent e in A we denote by Ne the ideal N Π Ae.We discuss first semi-primary rings A with radical N of square zero for which every simple module that is not a projective module is an injective module.We shall study the nonsemi-simple case, i.e., NΦO.Under this assumption JV becomes naturally a Γ-module.Let e, e' be primitive idempotents in A for which eNe' Φ 0. In particular Ne' Φ 0* From the exact sequence 0->Ne'-*Ae' ->S'->0, it follows that S' is not a projective module since Ae' is indecomposable.Since S' is a simple module it follows that S' is an injective module.Next consider the simple module Ae/Ne = S. Since eNe' Φ 0, since Ne' is a Γ-module, and since on N the Γ-module structure and the Λ-module structure coincide, Ne' contains a direct summand isomorphic with S. This gives rise to an exact sequence 0->S->Λe'~*K-»0 with K Φ 0. If S were injective this sequence would split, and this contradicts the indecomposability of Ae'.Therefore S is a projective module.Hence Ne' is a direct sum of projective modules, therefore Ne' is a projective module.The exact sequence 0-> Ne'-> Ae! -> S'->0 now implies ϊ.p.dim S' ^ 1, and since S' is not a projective module, then ϊ.p.dim S' = 1.Hence ϊ.p.dim^ Γ = 1, and this implies that A is an hereditary ring (i.e., l.gl.άimA = 1) [1].Conversely, assume that ί.gl.dimA = 1.Every ideal in A is the direct sum of N u -•-,N t where N λ is contained in the radical, and