global dimension

For any ring R, the left global dimension of R is defined to be the supremum of projective dimensions of left modules of R:

l.Gd(R):=sup{pdR(M)M is a left R-module }.

Similarly, the right global dimension of R is:

r.Gd(R):=sup{pdR(M)M is a right R-module }.

If R is commutative, then l.Gd(R)=r.Gd(R) and we may drop l and r and simply use Gd(R) to mean the global dimension of R.


  1. 1.

    For a ring R, l.Gd(R)=0 iff r.Gd(R)=0 (see the first example below). However, in general, l.Gd(R) is not necessarily the same as r.Gd(R).

  2. 2.

    The left (right) global dimension of a ring can also be defined in terms of injective dimensions. For example, for right global dimension of R, we have: r.Gd(R)=sup{idR(M)M is a right R-module }. This definition turns out to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the one using projective dimensions.


  1. 1.

    l.Gd(R)=0 iff R is a semisimple ringPlanetmathPlanetmath iff r.Gd(R)=0.

  2. 2.

    r.Gd(R)=1 iff R is a right hereditary ring that is not semisimplePlanetmathPlanetmathPlanetmath.

Title global dimension
Canonical name GlobalDimension
Date of creation 2013-03-22 14:51:51
Last modified on 2013-03-22 14:51:51
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 13D05
Classification msc 16E10
Classification msc 18G20
Synonym homological dimension