Commutative algebra | Ideals (ring theory)

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 (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). (Wikipedia).

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Related pages

Radical of an ideal | Local ring | Mathematics | Minimal prime ideal | Krull dimension | Maximal ideal | Noetherian ring | Length of a module | Primary ideal