On radical square zero rings

Abstract

Let \(\Lambda\) be a connected left artinian ring with radical square zero and with \(n\) simple modules. If \(\Lambda\)  is not self-injective, then we show that any module \(M\) with \(\operatorname{Ext}^i(M,\Lambda) = 0\) for \(1 \le i \le n+1\) is projective. We also determine the structure of the artin algebras with radical square zero and \(n\) simple modules which have a non-projective module \(M\) such that \(\operatorname{Ext}^i(M,\Lambda) = 0\) for \(1 \le i \le n\).