Nakayama algebra
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In mathematics, 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 {{harvs|txt|first=Tadasi |last=Nakayama|authorlink=Tadasi Nakayama|year=1940}} who called them "generalized uni-serial rings". These algebras were further studied by {{harvs|txt|first=Herbert|last=Kupisch|authorlink=Herbert Kupisch|year=1959}} and later by {{harvs|txt|first=Ichiro|last=Murase|year=1963-64}}, by {{harvs|txt|first=Kent Ralph|last=Fuller|year=1968}} and by {{harvs|txt|first=Idun|last=Reiten|year=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
- {{Citation | last1=Nakayama | first1=Tadasi | title=Note on uni-serial and generalized uni-serial rings | url=http://projecteuclid.org/euclid.pja/1195579089 | mr=0003618 | year=1940 | journal=Proc. Imp. Acad. Tokyo | volume=16 | pages=285–289}}
- {{Citation |last1=Fuller|first1=Kent Ralph|title=Generalized Uniserial Rings and their Kupisch Series|year=1968|journal= Math. Z.|volume=106|issue=4|pages=248–260|doi=10.1007/BF01110273|s2cid=122522745}}
- {{Citation|last1=Kupisch|first1=Herbert|title=Beiträge zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung|year=1959|journal=Crelle's Journal |volume=201|pages=100–112|doi=10.1515/crll.1959.201.100 }}
- {{Citation|last1=Murase|first1=Ichiro|title=On the structure of generalized uniserial rings III.|year=1964|journal=Sci. Pap. Coll. Gen. Educ., Univ. Tokyo|volume=14|pages=11–25}}
- {{Citation | last1=Reiten | first1=Idun | title=Representations of algebras (Puebla, 1980) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0094057 | mr=672115 | year=1982 | volume=944 | chapter=The use of almost split sequences in the representation theory of Artin algebras | pages=29–104| isbn=978-3-540-11577-9 }}