# 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 Tor*R**n*(*M*,*N*) 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 Cartan and Eilenberg (1956, p.122). 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 Tor*R**n*(*M*,*N*) 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*.

## Examples

- The module
**Q**of rational numbers over the ring**Z**of integers has weak dimension 0, but projective dimension 1. - The module
**Q**/**Z**over the ring**Z**has weak dimension 1, but injective dimension 0. - The module
**Z**over the ring**Z**has weak dimension 0, but injective dimension 1. - A Prüfer domain has weak global dimension at most 1.
- A Von Neumann regular ring has weak global dimension 0.
- A product of infinitely many fields has weak global dimension 0 but its global dimension is nonzero.
- If a ring is right Noetherian, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same.
- The triangular matrix ring has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian but not left Noetherian.

## References

- Cartan, Henri; Eilenberg, Samuel (1956),
*Homological algebra*, Princeton Mathematical Series,**19**, Princeton University Press, ISBN 978-0-691-04991-5, MR 0077480 - Năstăsescu, Constantin; Van Oystaeyen, Freddy (1987),
*Dimensions of ring theory*, Mathematics and its Applications,**36**, D. Reidel Publishing Co., doi:10.1007/978-94-009-3835-9, ISBN 9789027724618, MR 894033