A note on injectivity of monomial algebras

We show that a monomial algebra $\mathrm{\Lambda }$ over an algebraically closed field $K$ is self-injective if and only if each map $\mathrm{s}\mathrm{o}\mathrm{c}{(}_{\mathrm{\Lambda }}\mathrm{\Lambda })\to {}_{\mathrm{\Lambda }}\mathrm{\Lambda }$ can be extended to an endomorphism of ${}_{\mathrm{\Lambda }}\mathrm{\Lambda }$, 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.