Nakayama's conjecture

In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). (Leuschke Huneke) proved some cases of the generalized Nakayama conjecture. Nakayama's conjecture states that if all the modules of a minimal injective resolution of an Artin algebra R are injective and projective, then R is self-injective.

References

• Auslander, Maurice; Reiten, Idun (1975), "On a generalized version of the Nakayama conjecture", Proceedings of the American Mathematical Society 52 (1): 69–74, doi:10.2307/2040102, ISSN 0002-9939 
• Leuschke, Graham J.; Huneke, Craig (2004), "On a conjecture of Auslander and Reiten", Journal of Algebra 275 (2): 781–790, doi:10.1016/j.jalgebra.2003.07.018, ISSN 0021-8693 
• Nakayama, Tadasi (1958), "On algebras with complete homology", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 22: 300–307, doi:10.1007/BF02941960, ISSN 0025-5858