### 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 \(1 \le i \le n\).

#### Keywords

Artin algebras; left artinian rings; representations, modules; Gorenstein modules, CM modules; self-injective algebras; radical square zero algebras

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