Rings of finite representation type and modules of finite Morley rank
The paper under review is concerned with a model-theoretic and algebraic study of pure semisimple rings and rings of finite representation type. Recall that a ring R is right pure semisimple if each right R-module is a direct sum of finitely generated (indecomposable) modules. R is of finite representation type if moreover R has only a finite number of isomorphism classes of indecomposable modules. It is still an open question if these two notions coincide, equivalently if pure semi-simplicity is a left- right symmetric property for R, cf. \textit{D. Simson} [J. Algebra 71, 195- 218 (1981; Zbl 0477.16014]. Most of the characterizations of pure semisimplicity by properties of R-modules have their roots in a paper of \textit{L. Gruson} and \textit{C. U. Jensen} [C. R. Acad. Sci., Paris, Sér. A 282, 23-24 (1976; Zbl 0341.18010)]. Their results, centered around the notion of \(\Sigma\)-algebraic compact \((=\Sigma\)-pure-injective) modules and finitely definable subgroups, have been reproved and extended by various authors working either in algebra or model-theory. For a complete account we refer to the excellent bibliography of the present paper. Here we only mention the work of \textit{W. Zimmermann} [Commun. Algebra 5, 1083- 1117 (1977; Zbl 0371.16012)], \textit{D. Baer} [Arch. Math. 36, 495-501 (1981; Zbl 0441.16018)], both algebraic in spirit, and \textit{S. Garavaglia} [J. Symb. Logic 45, 155-164 (1980; Zbl 0453.03036)] and \textit{P. Rothmaler} [ibid. 48, 570-574 (1983; Zbl 0524.03018)] arguing from the view point of model theory and concentrating on the interpretation of \(\Sigma\)-algebraically compact modules as the totally transcendental modules. The interrelation stems mostly from the observation that the subgroups of finite definition are exactly the subgroups definable by positive primitive formulas. The author of the present paper does an excellent job in explaining the modeltheoretic background and the results in a language accessible to people working in algebra, and he succeeds in giving a unified presentation of the model-theoretic-algebraic characterizations of pure semisimple rings and rings of finite representation type. This takes nearly two third of his paper. The last section, using more advanced notions from model theory gives a characterization of rings of finite representation type by the property that all their modules have finite Morley rank. In the same spirit the author provides several characterizations of artinian rings.
\(\Sigma\)-algebraically compact modules
scientific article; zbMATH DE number 3855328
scientific article