TY - JOUR

T1 - Projectively stable artin algebras

T2 - Representation theory and quivers with relations

AU - Jagadeeshan, Shashidhar

AU - Kleiner, Mark

PY - 1999

Y1 - 1999

N2 - 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.

AB - 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.

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U2 - 10.1080/00927879908826565

DO - 10.1080/00927879908826565

M3 - Article

AN - SCOPUS:0033245506

SN - 0092-7872

VL - 27

SP - 2277

EP - 2316

JO - Communications in Algebra

JF - Communications in Algebra

IS - 5

ER -