System of parameters

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In mathematics, a system of parameters for a local Noetherian ring of Krull dimension d with maximal ideal m is a set of elements x₁, ..., x_d that satisfies any of the following equivalent conditions:

1. m is a minimal prime over (x₁, ..., x_d).
2. The radical of (x₁, ..., x_d) is m.
3. Some power of m is contained in (x₁, ..., x_d).
4. (x₁, ..., x_d) is m-primary.

Every local Noetherian ring admits a system of parameters.^[1]

It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would be less than d.

If M is a k-dimensional module over a local ring, then x₁, ..., x_k is a system of parameters for M if the length of M / (x₁, ..., x_k) M is finite.

• Atiyah, Michael Francis; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802

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