It is defined to be the supremum of the set of projective dimensions of all A-modules.
When the ring A is noncommutative, one initially has to consider two versions of this notion, right global dimension that arises from consideration of the right A-modules, and left global dimension that arises from consideration of the left A-modules.
Therefore, for noncommutative Noetherian rings, these two versions coincide and one is justified in talking about the global dimension.
[1] The right global dimension of a ring A can be alternatively defined as: The left global dimension of A has analogous characterizations obtained by replacing "right" with "left" in the above list.
This theorem opened the door to application of homological methods to commutative algebra.