In algebra, Auslander–Reiten theory studies the representation theory of Artinian rings using techniques such as Auslander–Reiten sequences (also called almost split sequences) and Auslander–Reiten quivers.
Auslander–Reiten theory was introduced by Maurice Auslander and Idun Reiten (1975) and developed by them in several subsequent papers.
Similarly for any finitely generated left module A that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism.
The module A in the almost split sequence is isomorphic to D Tr C, the dual of the transpose of C. Suppose that R is the ring k[x]/(xn) for a field k and an integer n≥1.
It has a map τ = D Tr called the translation from the non-projective vertices to the non-injective vertices, where D is the dual and Tr the transpose.