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 $R$:

$$l.\mathrm{Gd}(R):=\sup \{{\mathrm{pd}}_R(M)\mid M\text{ is a left R-module }\}.$$

Similarly, the right global dimension of $R$ is:

$$r.\mathrm{Gd}(R):=\sup \{{\mathrm{pd}}_R(M)\mid M\text{ is a right R-module }\}.$$

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: $r.\mathrm{Gd}(R)=\sup \{{\mathrm{id}}_R(M)\mid M\text{ is a right R-module }\}$. 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.