A note on injectivity of monomial algebras

Abstract

We show that a monomial algebra $\Lambda$ over an algebraically closed field $K$ is self-injective if and only if each map $\mathrm{soc}{(}_{\Lambda }\Lambda )\to {}_{\Lambda }\Lambda$ can be extended to an endomorphism of ${}_{\Lambda }\Lambda$, and provide a complete classification of such algebras. As a consequence, we show that the class of self-injective monomial algebras is a subclass of Nakayama algebras.