Weak dimension explained

R(M,N)
\operatorname{Tor}
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 . 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

R(M,N)
\operatorname{Tor}
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

\Q

of rational numbers over the ring

\Z

of integers has weak dimension 0, but projective dimension 1.

\Q/\Z

over the ring

\Z

has weak dimension 1, but injective dimension 0.

\Z

over the ring

\Z

has weak dimension 0, but injective dimension 1.

\begin{bmatrix}\Z&\Q\\0&\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