On modules with the Kulikov property and pure semisimple modules and rings (Q1814226)

A ring is called right pure-semisimple if every right module is a direct sum of finitely generated submodules. It has been independently proved by \textit{M. Prest} [J. Lond. Math. Soc., II. Ser. 38, 403-409 (1988; Zbl 0674.16019)] and by \textit{B. Zimmermann-Huisgen} and \textit{W. Zimmermann} [Trans. Am. Math. Soc. 320, 695-711 (1990; Zbl 0699.16019)] that over such a ring for every \(n\in\mathbb{N}\) up to isomorphism there are only finitely many indecomposable right modules of length \(\leq n\). As for the proofs, the first mentioned paper rests on methods of model theory, whereas in the second one only module theory is applied. In the present article a further proof is offered, using properties of functor categories, in particular results by \textit{H. Brune} [J. Pure Appl. Algebra 28, 31-39 (1983; Zbl 0507.16030)].