The dual transpose over an artin algebra Λ and over a factor of Λ

Let Λ be an artin algebra, a two-sided ideal of Λ, and let M be an indecomposable nonprojective -module. We consider first two particular embeddingsfAR,fAZ from into τΛM defined, respectively, by Auslander–Reiten and Assem–Zacharia in case Λ is a split by nilpotent extension of by . We prove that fAR and fAZ coincide in the sense that there exists a Λ-isomorphism such that fAZ=fAR . Secondly, we give some relationships between the dual transpose over Λ and over a factor of Λ and applications.