Generalized local ring: Difference between revisions
(New page: ==Definition== ===Symbol-free definition=== A positively graded commutative unital ring ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local...) |
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A [[graded ring|positively graded]] [[commutative unital ring]] ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local ring|local]] and [[Noetherian ring|Noetherian]], and ''R'' is a finitely generated as an algebra over its degree 0 part. | A [[graded ring|positively graded]] [[commutative unital ring]] ''R'' is termed a '''generalized local ring''' if its degree 0 part is [[local ring|local]] and [[Noetherian ring|Noetherian]], and ''R'' is a finitely generated as an algebra over its degree 0 part. | ||
==Metaproperties== | |||
==Uniqueness of maximal ideal== | ===Uniqueness of maximal ideal=== | ||
A generalized local ring has a unique maximal homogeneous ideal. | A generalized local ring has a unique maximal homogeneous ideal. | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[ | * [[Multivariate polynomial ring over a field]] | ||
Latest revision as of 19:14, 3 January 2009
Definition
Symbol-free definition
A positively graded commutative unital ring R is termed a generalized local ring if its degree 0 part is local and Noetherian, and R is a finitely generated as an algebra over its degree 0 part.
Metaproperties
Uniqueness of maximal ideal
A generalized local ring has a unique maximal homogeneous ideal.