Commutative algebra | Homological algebra | Ring theory

Weak dimension

In abstract algebra, the weak dimension of a nonzero right module M over a ring R is the largest number n such that the Tor group is nonzero for some left R-module N (or infinity if no largest such n exists), and the weak dimension of a left R-module is defined similarly. The weak dimension was introduced by Henri Cartan and Samuel Eilenberg . The weak dimension is sometimes called the flat dimension as it is the shortest length of a resolution of the module by flat modules. The weak dimension of a module is at most equal to its projective dimension. The weak global dimension of a ring is the largest number n such that is nonzero for some right R-module M and left R-module N. If there is no such largest number n, the weak global dimension is defined to be infinite. It is at most equal to the left or right global dimension of the ring R. (Wikipedia).

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Related pages

Global dimension | Resolution (algebra) | Flat module | Abstract algebra | Von Neumann regular ring | Rational number | Integer | Field (mathematics) | Product of rings | Prüfer domain | Noetherian ring | Tor functor | Triangular matrix ring | Ring (mathematics) | Module (mathematics)