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Abelian groups like modules. (English) Zbl 0628.16014
The author studies modules \(M_ R\) satisfying the following condition: (I) every finitely generated submodule of a homomorphic image of \(M_ R\) is a direct sum of uniserial modules \((M_ R\) is said to be uniserial if \(M_ R\) has finite composition length and the lattice of its submodules is linearly ordered under inclusion). The modules satisfying condition (I) are called QTAG-modules.
Following R. B. Warfield jun. [see J. Algebra 37, 187-222 (1975; Zbl 0319.16025)] a ring R is said to be right (left) serial if \(R_ R\) (respectively \({}_ RR)\) is a direct sum of modules with linearly ordered lattices of submodules. A ring R which is right as well as left, serial and artinian, is called a generalized uniserial ring.
The author develops the structure theory of QTAG-modules. He introduces the concepts of exponent and height for an element in a QTAG-module. It is shown that any QTAG-module M admits a basic submodule and that any two basic submodules of M are isomorphic. In the paper the rings R for which \(R_ R\) is a QTAG-module are studied.
The main results for these rings are given in the following theorems: Theorem 1. Let R be any indecomposable ring such that \(R_ R\) is a QTAG- module and for any indecomposable idempotent \(e\in R\), \(eJ^ 2\neq 0\) (J denotes Jacobson radical). Then either R is a local ring or R is generalized uniserial. Theorem 2. Let R be any non-local indecomposable ring such that \(R_ R\) is a QTAG-module. Then either R is a generalized uniserial ring or it has an indecomposable idempotent \(e\in R\) such that \(eJ^ 2\neq 0\) and \(eR/eJ^ 2\) is not quasi-injective.
Reviewer: I.Bekker

MSC:
16D80 Other classes of modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16Gxx Representation theory of associative rings and algebras
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
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