In
mathematics, 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
* Local government, a form of public administration, usually the lowest tier of administra ...
Noetherian ring of
Krull dimension ''d'' with
maximal ideal ''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 of .
General references
*
References
category:Commutative algebra
Ideals (ring theory)
{{algebra-stub