The global dimension of a ring is the supremum of the projective dimensions of all its modules, allowing infinity. It measures the longest projective resolution needed by any module. For a field it is zero, since every module is a vector space and hence free.
If global dimension is zero, every module is a projective module. Each short exact sequence splits because its quotient is projective. Conversely, splitting every short exact sequence makes each module a direct summand of a free module and therefore a projective module. Splitting also makes every module an injective module: for choose a retraction and compose it with any prescribed map out of .
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The term "global dimension" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Mathematics**: In category theory, the global dimension of a ring is a measure of how "complex" its modules are. It is defined as the supremum of the projective dimensions of all modules over the ring. A ring with finite global dimension has all its modules that can be resolved by a finite projective resolution.