In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a system of parameters for a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
Arts, entertainment, and media
* ''Local'' (comics), a limited series comic book by Bria ...
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
of
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
''d'' with
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
''m'' is a set of elements ''x''
1, ..., ''x''
''d'' that satisfies any of the following equivalent conditions:
# ''m'' is a
minimal prime over (''x''
1, ..., ''x''
''d'').
# The
radical of (''x''
1, ..., ''x''
''d'') is ''m''.
# Some power of ''m'' is contained in (''x''
1, ..., ''x''
''d'').
# (''x''
1, ..., ''x''
''d'') 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 ''x''
1, ..., ''x''
''k'' is a system of parameters for ''M'' if the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
of is finite.
General references
*
References
category:Commutative algebra
Ideals (ring theory)
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