# Weak dimension

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In [abstract algebra](/source/Abstract_algebra), the **weak dimension** of a [nonzero](/source/Zero_module) right [module](/source/Module_(mathematics)) *M* over a [ring](/source/Ring_(mathematics)) *R* is the largest number *n* such that the [Tor group](/source/Tor_functor) Tor n R ⁡ ( M , N ) {\displaystyle \operatorname {Tor} _{n}^{R}(M,N)} is [nonzero](/source/Zero_group) 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](/source/Henri_Cartan) and [Samuel Eilenberg](/source/Samuel_Eilenberg) ([1956](#CITEREFCartanEilenberg1956), p.122). The weak dimension is sometimes called the **flat dimension** as it is the shortest length of the [resolution](/source/Resolution_(algebra)) of the module by [flat modules](/source/Flat_module). The weak dimension of a module is, at most, equal to its [projective dimension](/source/Projective_dimension).

The **weak global dimension** of a ring is the largest number *n* such that Tor n R ⁡ ( M , N ) {\displaystyle \operatorname {Tor} _{n}^{R}(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](/source/Global_dimension) of the ring *R*.

## Examples

- The module Q {\displaystyle \mathbb {Q} } of [rational numbers](/source/Rational_number) over the ring Z {\displaystyle \mathbb {Z} } of [integers](/source/Integer) has weak dimension 0, but projective dimension 1.

- The module Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } over the ring Z {\displaystyle \mathbb {Z} } has weak dimension 1, but [injective dimension](/source/Injective_dimension) 0.

- The module Z {\displaystyle \mathbb {Z} } over the ring Z {\displaystyle \mathbb {Z} } has weak dimension 0, but injective dimension 1.

- A [Prüfer domain](/source/Pr%C3%BCfer_domain) has weak global dimension at most 1.

- A [Von Neumann regular ring](/source/Von_Neumann_regular_ring) has weak global dimension 0.

- A [product](/source/Product_of_rings) of infinitely many [fields](/source/Field_(mathematics)) has weak global dimension 0 but its global dimension is nonzero.

- If a ring is right [Noetherian](/source/Noetherian_ring), 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](/source/Triangular_matrix_ring) [ Z Q 0 Q ] {\displaystyle {\begin{bmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{bmatrix}}} 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](/source/Henri_Cartan); [Eilenberg, Samuel](/source/Samuel_Eilenberg) (1956), [*Homological algebra*](https://books.google.com/books?id=0268b52ghcsC), Princeton Mathematical Series, vol. 19, [Princeton University Press](/source/Princeton_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-691-04991-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-04991-5), [MR](/source/MR_(identifier)) [0077480](https://mathscinet.ams.org/mathscinet-getitem?mr=0077480) {{[citation](https://en.wikipedia.org/wiki/Template:Citation)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

- Năstăsescu, Constantin; [Van Oystaeyen, Freddy](/source/Fred_Van_Oystaeyen) (1987), *Dimensions of ring theory*, Mathematics and its Applications, vol. 36, D. Reidel Publishing Co., [doi](/source/Doi_(identifier)):[10.1007/978-94-009-3835-9](https://doi.org/10.1007%2F978-94-009-3835-9), [ISBN](/source/ISBN_(identifier)) [9789027724618](https://en.wikipedia.org/wiki/Special:BookSources/9789027724618), [MR](/source/MR_(identifier)) [0894033](https://mathscinet.ams.org/mathscinet-getitem?mr=0894033)

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