Buchsbaum ring

In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence ${\displaystyle (a_1,\cdots ,a_r)}$ of the maximal ideal ${\displaystyle m}$ is called a weak sequence if ${\displaystyle m\cdot ((a_1,\cdots ,a_{i-1}):a_i)\subset (a_1,\cdots ,a_{i-1})}$ for all ${\displaystyle i}$.

They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973) and are named after David Buchsbaum.

Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring.

References

• Buchsbaum, D. (1966), "Complexes in local ring theory", in Herstein, I. N., Some aspects of ring theory, Centro Internazionale Matematico Estivo (C.I.M.E.). II Ciclo, Varenna (Como), 23-31 agosto, 1965, Rome: Edizioni cremonese, pp. 223–228, ISBN 978-3-642-11035-1, https://books.google.com/books?id=5mBgpQI3aekC 
• Hazewinkel, Michiel, ed. (2001), "Buchsbaum ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Stückrad, Jürgen; Vogel, Wolfgang (1973), "Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie", Journal of Mathematics of Kyoto University 13: 513–528, ISSN 0023-608X, http://projecteuclid.org/euclid.kjm/1250523322 
• Stückrad, Jürgen; Vogel, Wolfgang (1986), Buchsbaum rings and applications, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16844-7, https://books.google.com/books?id=xBTvAAAAMAAJ 

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