weak global dimension

Let R be a ring. The (right) weak global dimension of R is defined as

w.gl.dimR=sup{wdRM|M is a right module}.

Unlike global dimension of R the definition of the weak global dimension is left/right symmetricPlanetmathPlanetmath. This follows from the fact that for every left module M and right module N there is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


Thus we simply say that R has the weak global dimension. Note that this does not hold for Ext functors, so (generally) the definition of global dimension is not left/right symmetric.

The following propositionPlanetmathPlanetmathPlanetmath is a simple consequence of the fact that every projective moduleMathworldPlanetmath is flat:

Proposition. For any ring R we have


where l.gl.dim and r.gl.dim denote the left global dimension and right global dimension respectively.

Title weak global dimension
Canonical name WeakGlobalDimension
Date of creation 2013-03-22 19:18:42
Last modified on 2013-03-22 19:18:42
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 16E05