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System of Parameters

A system of parameters of a commutative Noetherian local ring R of Krull dimension d is a sequence x_1,...,x_d in its maximal ideal m such that the quotient ring R/<x_1,...,x_d> has Krull dimension zero (Bruns and Herzog 1998). Equivalently, the ideal radical of the generated ideal is m.

A system of parameters need not be a ring regular sequence. Every system of parameters of a Cohen-Macaulay ring is a ring regular sequence. For example, x_1,...,x_d is a system of parameters of the power series ring k[[x_1,...,x_d]] over a field k; its quotient ring is k.

See also

Big Cohen-Macaulay Module, Cohen-Macaulay Ring, Krull Dimension, Local Ring, Ring Regular Sequence

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.

Cite this as:

Weisstein, Eric W. "System of Parameters." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SystemofParameters.html

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