Abstract
Let A be an artin algebra over a commutative artin ring R and mod A be the category of finitely generated right A-modules. A cycle in mod A is a sequence of non-zero non-isomorphisms M 0 → M 1 → ... → M n = M 0 between indecomposable modules from mod A. The main aim of this survey article is to show that study of cycles in mod A leads to interesting information on indecomposable A-modules, the Auslander-Reiten quiver of A, and the ring structure of A. We present recent advances in some areas of the representation theory of artin algebras which should be of interest to a wider audience. In the paper, we also pose a number of open problems and indicate some new research directions.
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