Modules of finite length over their endomorphism rings.

*(English)*Zbl 0805.16028
Representations of algebras and related topics, Proc. Tsukuba Int. Conf., Kyoto/Jap. 1990, Lond. Math. Soc. Lect. Note Ser. 168, 127-184 (1992).

[For the entire collection see Zbl 0746.00076.]

Given a ring \(R\) (associative, with 1) one can define the endolength of an \(R\)-module \(M\) to be its length when it is regarded in the natural way as an \(\text{End}_ R(M)\)-module, and thus one can consider the class of modules of finite endolength. The aim of this paper is to show that this is a useful concept. Briefly, the contents are as follows. In §§1-3 we cover some background machinery, in §§4-6 we discuss the modules of finite endolength for a general ring, and in §§7-9 we show how these modules control the behaviour of the finite length modules for noetherian and Artin algebras. Although much of this paper has a survey nature, there are some new results proved here, the main ones being the characterization of the pure-injective modules which occur as the source of a left almost split map in §2, the character theory for modules of finite endolength in §5, and the characterization of the Artin algebras with an indecomposable module of infinite length and finite endolength (a generic module) proved in §§8-9.

Given a ring \(R\) (associative, with 1) one can define the endolength of an \(R\)-module \(M\) to be its length when it is regarded in the natural way as an \(\text{End}_ R(M)\)-module, and thus one can consider the class of modules of finite endolength. The aim of this paper is to show that this is a useful concept. Briefly, the contents are as follows. In §§1-3 we cover some background machinery, in §§4-6 we discuss the modules of finite endolength for a general ring, and in §§7-9 we show how these modules control the behaviour of the finite length modules for noetherian and Artin algebras. Although much of this paper has a survey nature, there are some new results proved here, the main ones being the characterization of the pure-injective modules which occur as the source of a left almost split map in §2, the character theory for modules of finite endolength in §5, and the characterization of the Artin algebras with an indecomposable module of infinite length and finite endolength (a generic module) proved in §§8-9.

##### MSC:

16S50 | Endomorphism rings; matrix rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16D50 | Injective modules, self-injective associative rings |

16G10 | Representations of associative Artinian rings |

16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |

16P20 | Artinian rings and modules (associative rings and algebras) |

16P40 | Noetherian rings and modules (associative rings and algebras) |