Examples
Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem. More generally, if R is a Noetherian ring of finite global dimension k and A = R is a ring of polynomials in one variable over R then the global dimension of A is equal to k + 1.
The first Weyl algebra A1 is a noncommutative Noetherian domain of global dimension one.
A ring has global dimension zero if and only if it is semisimple. The global dimension of a ring A is less than or equal to one if and only if A is hereditary. In particular, a commutative principal ideal domain which is not a field has global dimension one.
Read more about this topic: Global Dimension
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