{{No footnotes|date=June 2021}} In [[abstract algebra]], the '''weak dimension''' of a [[zero module|nonzero]] right [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is the largest number ''n'' such that the [[Tor functor|Tor group]] <math>\operatorname{Tor}_n^R(M,N)</math> is [[zero group|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 {{harvs|txt|last1=Cartan | first1=Henri | author1-link= Henri Cartan | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg |year=1956|loc=p.122}}. The weak dimension is sometimes called the '''flat dimension''' as it is the shortest length of the [[Resolution (algebra)|resolution]] of the module by [[flat module]]s. 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 <math>\operatorname{Tor}_n^R(M,N)</math> 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 <math>\Q</math> of [[rational number]]s over the ring <math>\Z</math> of [[integer]]s has weak dimension 0, but projective dimension 1. *The module <math>\Q/\Z</math> over the ring <math>\Z</math> has weak dimension 1, but [[injective dimension]] 0. *The module <math>\Z</math> over the ring <math>\Z</math> 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 rings|product]] of infinitely many [[field (mathematics)|fields]] has weak global dimension 0 but its global dimension is nonzero. *If a ring is right [[Noetherian ring|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]] <math>\begin{bmatrix}\Z&\Q \\0&\Q \end{bmatrix}</math> has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian, but not left Noetherian.
==References==
*{{Citation | last1=Cartan | first1=Henri | author1-link= Henri Cartan | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Homological algebra | url=https://books.google.com/books?id=0268b52ghcsC | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | isbn=978-0-691-04991-5 |mr=0077480 | year=1956 | volume=19}} *{{Citation | last1=Năstăsescu | first1=Constantin | last2=Van Oystaeyen | first2=Freddy | author2-link = Fred Van Oystaeyen | title=Dimensions of ring theory | publisher=D. Reidel Publishing Co. | series=Mathematics and its Applications | isbn=9789027724618 | doi=10.1007/978-94-009-3835-9 |mr=894033 | year=1987 | volume=36}}
[[Category:Commutative algebra]] [[Category:Ring theory]] [[Category:Homological algebra]]
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