# Buchsbaum ring

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In [mathematics](/source/Mathematics), **Buchsbaum rings** are Noetherian [local rings](/source/Local_ring) such that every [system of parameters](/source/System_of_parameters) is a weak sequence. A sequence ( a 1 , ⋯ , a r ) {\displaystyle (a_{1},\cdots ,a_{r})} of the maximal ideal m {\displaystyle m} is called a weak sequence if m ⋅ ( ( a 1 , ⋯ , a i − 1 ) : a i ) ⊂ ( a 1 , ⋯ , a i − 1 ) {\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})} for all i {\displaystyle i} .

They were introduced by Jürgen Stückrad and Wolfgang Vogel ([1973](#CITEREFStückradVogel1973)) and are named after [David Buchsbaum](/source/David_Buchsbaum).

Every [Cohen–Macaulay local ring](/source/Cohen%E2%80%93Macaulay_ring) is a Buchsbaum ring. Every Buchsbaum ring is a [generalized Cohen–Macaulay ring](/source/Generalized_Cohen%E2%80%93Macaulay_ring).

## References

- Buchsbaum, D. (1966), "Complexes in local ring theory", in [Herstein, I. N.](/source/Israel_Nathan_Herstein) (ed.), [*Some aspects of ring theory*](https://books.google.com/books?id=5mBgpQI3aekC), Centro Internazionale Matematico Estivo (C.I.M.E.). II Ciclo, Varenna (Como), 23-31 agosto, vol. 1965, Rome: Edizioni cremonese, pp. 223–228, [ISBN](/source/ISBN_(identifier)) [978-3-642-11035-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-11035-1), [MR](/source/MR_(identifier)) [0223393](https://mathscinet.ams.org/mathscinet-getitem?mr=0223393)

- Goto, Shiro (2001) [1994], ["Buchsbaum ring"](https://www.encyclopediaofmath.org/index.php?title=Buchsbaum_ring), *[Encyclopedia of Mathematics](/source/Encyclopedia_of_Mathematics)*, [EMS Press](/source/European_Mathematical_Society)

- Stückrad, Jürgen; Vogel, Wolfgang (1973), ["Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie"](http://projecteuclid.org/euclid.kjm/1250523322), *Journal of Mathematics of Kyoto University*, **13**: 513–528, [ISSN](/source/ISSN_(identifier)) [0023-608X](https://search.worldcat.org/issn/0023-608X), [MR](/source/MR_(identifier)) [0335504](https://mathscinet.ams.org/mathscinet-getitem?mr=0335504)

- Stückrad, Jürgen; Vogel, Wolfgang (1986), [*Buchsbaum rings and applications*](https://books.google.com/books?id=xBTvAAAAMAAJ), Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-3-540-16844-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-16844-7), [MR](/source/MR_(identifier)) [0881220](https://mathscinet.ams.org/mathscinet-getitem?mr=0881220)

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Adapted from the Wikipedia article [Buchsbaum ring](https://en.wikipedia.org/wiki/Buchsbaum_ring) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Buchsbaum_ring?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.