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 $R$:
$$l.\mathrm{Gd}(R):=\mathrm{sup}\{{\mathrm{pd}}_{R}(M)\mid M\text{is a left Rmodule}\}.$$ 
Similarly, the right global dimension of $R$ is:
$$r.\mathrm{Gd}(R):=\mathrm{sup}\{{\mathrm{pd}}_{R}(M)\mid M\text{is a right Rmodule}\}.$$ 
If $R$ is commutative, then $l.\mathrm{Gd}(R)=r.\mathrm{Gd}(R)$ and we may drop $l$ and $r$ and simply use $\mathrm{Gd}(R)$ to mean the global dimension of $R$.
Remarks.

1.
For a ring $R$, $l.\mathrm{Gd}(R)=0$ iff $r.\mathrm{Gd}(R)=0$ (see the first example below). However, in general, $l.\mathrm{Gd}(R)$ is not necessarily the same as $r.\mathrm{Gd}(R)$.

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: $r.\mathrm{Gd}(R)=\mathrm{sup}\{{\mathrm{id}}_{R}(M)\mid M\text{is a right Rmodule}\}$. This definition turns out to be equivalent^{} to the one using projective dimensions.
Examples.

1.
$l.\mathrm{Gd}(R)=0$ iff $R$ is a semisimple ring^{} iff $r.\mathrm{Gd}(R)=0$.

2.
$r.\mathrm{Gd}(R)=1$ iff $R$ is a right hereditary ring that is not semisimple^{}.
Title  global dimension 

Canonical name  GlobalDimension 
Date of creation  20130322 14:51:51 
Last modified on  20130322 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 