Weak dimension

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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 ${\displaystyle \operatorname {Tor} _{n}^{R}(M,N)}$ 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 Henri Cartan and Samuel Eilenberg (1956, p.122). The weak dimension is sometimes called the flat dimension as it is the shortest length of the 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 ${\displaystyle \operatorname {Tor} _{n}^{R}(M,N)}$ 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 of the ring R.