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- Global_dimension abstract "In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings. By a theorem of Jean-Pierre Serre, global dimension can be used to characterize within the class of commutative Noetherian local rings those rings which are regular. Their global dimension coincides with the Krull dimension, whose definition is module-theoretic.When the ring A is noncommutative, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right A-modules, and left global dimension that arises from consideration of the left A-modules. For an arbitrary ring A the right and left global dimensions may differ. However, if A is a Noetherian ring, both of these dimensions turn out to be equal to weak global dimension, whose definition is left-right symmetric. Therefore, for noncommutative Noetherian rings, these two versions coincide and one is justified in talking about the global dimension.".
- Global_dimension wikiPageID "2291739".
- Global_dimension wikiPageRevisionID "544070309".
- Global_dimension hasPhotoCollection Global_dimension.
- Global_dimension subject Category:Dimension.
- Global_dimension subject Category:Homological_algebra.
- Global_dimension subject Category:Module_theory.
- Global_dimension subject Category:Ring_theory.
- Global_dimension comment "In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings.".
- Global_dimension label "Global dimension".
- Global_dimension label "內射維度、投射維度與同調維度".
- Global_dimension sameAs m.071byk.
- Global_dimension sameAs Q5570832.
- Global_dimension sameAs Q5570832.
- Global_dimension wasDerivedFrom Global_dimension?oldid=544070309.
- Global_dimension isPrimaryTopicOf Global_dimension.