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: Every local Noetherian ring admits a system of parameters.
 * 1) m is a minimal prime over (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.

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.