System of parameters
In commutative algebra, a system of parameters for a local ring of Krull dimension d with maximal ideal m is a set of elements x1, ..., xd that satisfies any of the following equivalent conditions:
- m is a minimal prime of (x1, ..., xd).
- The radical of (x1, ..., xd) is m.
- Some power of m is contained in (x1, ..., xd).
- (x1, ..., xd) is m-primary.
Every local Noetherian ring admits a system of parameters.
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, ..., xk is a system of parameters for M if the length of M / (x1, ..., xk)M is finite.
References
- 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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