Extension problem associated with socle of monomial algebras.

Abstract

We show that a monomial algebra Λ over an algebraically closed field K is self-injective if and only if each map soc(ΛΛ) −→ ΛΛ can be extended to an endomorphism of ΛΛ, 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.