System of parameters
Encyclopedia
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:
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.
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, a system of parameters for a local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
of Krull dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
d with maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
m is a set of elements x1, ..., xd that satisfies any of the following equivalent conditions:
- m is a minimal prime of (x1, ..., xd).
- The radicalRadical of an idealIn commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
of (x1, ..., xd) is m. - Some power of m is contained in (x1, ..., xd).
- (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 is finite.