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.