System of parameters

In commutative algebra, a system of parameters for a local ring of Krull dimension n with maximal ideal m is a set of elements x₁, ..., x_n that satisfies any of the following equivalent conditions:

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

Every local Noetherian ring admits a system of parameters.

It is not possible for fewer than n elements to generate an ideal whose radical is m because then the dimension of R would be less than n.

If M is a module over a local ring, then x₁, ..., x_d is a system of parameters for M if the length of M / (x₁, ..., x_d)M is finite.