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:

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

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