Taiwanese Journal of Mathematics

Rings with Indecomposable Right Modules Local

Surjeet Singh

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Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

Article information

Taiwanese J. Math., Volume 14, Number 6 (2010), 2261-2275.

First available in Project Euclid: 18 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16G10: Representations of Artinian rings 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander- Reiten quivers

left serial rings generalized uniserial rings exceptional rings rings of finite representation type $M$-injective modules and $M$-projective modules


Singh, Surjeet. Rings with Indecomposable Right Modules Local. Taiwanese J. Math. 14 (2010), no. 6, 2261--2275. doi:10.11650/twjm/1500406074. https://projecteuclid.org/euclid.twjm/1500406074

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