### Abstract

A left artinian ring A is projectively stable if no non-zero morphism f : M → N of finitely generated left A-modules M, N having no non-zero projective direct summands factors through a projective module. Such rings have many good properties and, although the definition is given in terms of the category of modules, the structure of projectively stable left artinian rings allows a satisfying description: they are "built" from well-known left artinian rings, left hereditary and serial, by a pullback of rings construction. We show that representations of projectively stable artin algebras are also "built" from representations of well-studied artin algebras, hereditary and serial. Namely, we give a simple geometric construction that produces the Auslander-Reiten quiver or the ordinary quiver of a projectively stable algebra from those of an hereditary algebra and a serial algebra.

Original language | English (US) |
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Pages (from-to) | 2277-2316 |

Number of pages | 40 |

Journal | Communications in Algebra |

Volume | 27 |

Issue number | 5 |

DOIs | |

State | Published - 1999 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Communications in Algebra*,

*27*(5), 2277-2316. https://doi.org/10.1080/00927879908826565