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 TorR
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 CartanandEilenberg (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.