Abstract

<jats:p>Let <jats:italic>A</jats:italic> be an artin algebra over a commutative artin ring <jats:italic>R</jats:italic>, mod <jats:italic>A</jats:italic> be the category of finitely generated right <jats:italic>A</jats:italic>-modules, and rad<jats:sup>∞</jats:sup> (mod<jats:italic>A</jats:italic>) be the infinite power of the Jacobson radical rad(mod<jats:italic>A</jats:italic>) of mod<jats:italic>A</jats:italic>. Recall that <jats:italic>A</jats:italic> is said to be representation-finite if mod <jats:italic>A</jats:italic> admits only finitely many non-isomorphic indecomposable modules. It is known that <jats:italic>A</jats:italic> is representation-finite if and only if rad<jats:sup>∞</jats:sup> (mod <jats:italic>A</jats:italic>) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [<jats:bold>26</jats:bold>, <jats:bold>2</jats:bold>] we know that <jats:italic>A</jats:italic> is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod <jats:italic>A</jats:italic>.</jats:p>