Nakayama algebra

In algebra, a Nakayama algebra or generalized uniserial algebra is an algebra such that each left or right indecomposable projective module has a unique composition series. They were studied by Tadasi Nakayama (1940) who called them "generalized uni-serial rings". These algebras were further studied by Herbert Kupisch (1959) and later by Ichiro Murase (1963-64), by Kent Ralph Fuller (1968) and by Idun Reiten (1982).

An example of a Nakayama algebra is k[x]/(xⁿ) for k a field and n a positive integer.

Current usage of uniserial differs slightly: an explanation of the difference appears here.

References

• Nakayama, Tadasi (1940), "Note on uni-serial and generalized uni-serial rings", Proc. Imp. Acad. Tokyo, 16: 285–289, MR 0003618
• Fuller, Kent Ralph (1968), "Generalized Uniserial Rings and their Kupisch Series", Math. Z., 106 (4): 248–260, doi:10.1007/BF01110273, S2CID 122522745
• Kupisch, Herbert (1959), "Beiträge zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung", Crelle's Journal, 201: 100–112
• Murase, Ichiro (1964), "On the structure of generalized uniserial rings III.", Sci. Pap. Coll. Gen. Educ., Univ. Tokyo, 14: 11–25
• Reiten, Idun (1982), "The use of almost split sequences in the representation theory of Artin algebras", Representations of algebras (Puebla, 1980), Lecture Notes in Mathematics, vol. 944, Berlin, New York: Springer-Verlag, pp. 29–104, doi:10.1007/BFb0094057, ISBN 978-3-540-11577-9, MR 0672115