A note on injectivity of monomial algebras

TL;DR

This paper characterizes when monomial algebras are self-injective over an algebraically closed field, providing a classification and linking them to Nakayama algebras.

Contribution

It offers a necessary and sufficient condition for self-injectivity of monomial algebras and classifies all such algebras.

Findings

Self-injective monomial algebras are characterized by extension properties of their socle maps.

The class of self-injective monomial algebras is a subset of Nakayama algebras.

A complete classification of self-injective monomial algebras is provided.

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.