  We study persistence modules defined on commutative ladders. This class of
persistence modules frequently appears in topological data analysis, and the
theory and algorithm proposed in this paper can be applied to these practical
problems. A new algebraic framework deals with persistence modules as
representations on associative algebras and the Auslander-Reiten theory is
applied to develop the theoretical and algorithmic foundations. In particular,
we prove that the commutative ladders of length less than 5 are
representation-finite and explicitly show their Auslander-Reiten quivers.
Furthermore, a generalization of persistence diagrams is introduced by using
Auslander-Reiten quivers. We provide an algorithm for computing persistence
diagrams for the commutative ladders of length 3 by using the structure of
Auslander-Reiten quivers.
