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 x1, ..., xd that satisfies any of the following equivalent conditions: m is a minimal prime over (x1, ..., xd). The radical of (x1, ..., xd) is m. Some power of m is contained in (x1, ..., xd). (x1, ..., xd) is m-primary. R/(x1, ..., xd) is an Artinian ring. 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.
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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, ..., xd that satisfies any of the following equivalent conditions:
- m is a minimal prime over (x1, ..., xd).
- The radical of (x1, ..., xd) is m.
- Some power of m is contained in (x1, ..., xd).
- (x1, ..., xd) is m-primary.
- R/(x1, ..., xd) is an Artinian ring.
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, ..., xk is a system of parameters for M if the length of M / (x1, ..., xk) M is finite.
General references
[edit]- Atiyah, Michael Francis; Macdonald, I. G. (1969), Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0242802
- Eisenbud, David (1995), Commutative Algebra, Springer-Verlag, New York, MR 1322960
References
[edit]- ^ "Math 711: Lecture of September 5, 2007" (PDF). University of Michigan. September 5, 2007. Retrieved March 9, 2026.
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