Canonical Comultiplication and Double Centraliser Property

In this paper, we show the existence of the attached comultiplication structure on Hom(eAe)(eA, D(Ae)) if an (eAe, A)-bimodule eA has the double centraliser property over an algebra A with the idempotent e. Then we apply it on gendo-Gorenstein algebras. As applications, we give a sufficient and necessary condition for a gendo-Gorenstein algebra to be Gorenstein, and give a bocs-theoretic characterisation of the double centraliser property.