In mathematics, Buchsbaum rings are Noetherian

local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...

s such that every

system of parameters In mathematics, a system of parameters for a local Noetherian ring of Krull dimension ''d'' with maximal ideal ''m'' is a set of elements ''x''1, ..., ''x'd'' that satisfies any of the following equivalent conditions: # ''m'' is a minimal pri ...

is a weak sequence. A sequence

(a_1,\cdots,a_r)

of the maximal ideal

m

is called a weak sequence if

m\cdot((a_1,\cdots,a_)\colon
a_i)\subset(a_1,\cdots,a_)

for all

i

. They were introduced by and are named after

David Buchsbaum David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory. He proved the Auslander–Buchsbaum formula and the Ausla ...

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

generalized Cohen–Macaulay ring In algebra, a generalized Cohen–Macaulay ring is a commutative Noetherian local ring (A, \mathfrak) of Krull dimension ''d'' > 0 that satisfies any of the following equivalent conditions: *For each integer i = 0, \dots, d - 1, the length of the ' ...

References

* * * * Commutative algebra Ring theory {{abstract-algebra-stub