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:
  1. m is a minimal prime over.
  2. The radical of is m.
  3. Some power of m is contained in.
  4. is m-primary.
  5. R/ 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 is finite.