The module depth of a nonzero finitely generated module over a Noetherian local ring is the largest length of a ring regular sequence for contained in its maximal ideal . Such a sequence , ..., has the property that multiplication by is injective on and the final quotient module is nonzero (Bruns and Herzog 1998, pp. 3 and 10).

The depth is at most the Krull dimension (Bruns and Herzog 1998, p. 12). For , it is the depth of the ring. A Cohen-Macaulay ring that is a local ring has depth equal to its Krull dimension (Stacks Project Authors 2026).

See also

Cohen-Macaulay Ring, Krull Dimension, Local Ring, Module, Ring Regular Sequence, Serre Condition S₂, Small Cohen-Macaulay Module

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.The Stacks Project Authors. "Cohen-Macaulay Rings." §10.104 in The Stacks Project, Tag 00N7, 2026. https://stacks.math.columbia.edu/tag/00N7.

Cite this as:

Weisstein, Eric W. "Module Depth." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ModuleDepth.html

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