• Let
A =
K[
x1,...,
xn] 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[x] is a ring of polynomials in one variable over
R then the global dimension of
A is equal to
k + 1. • 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. For example \mathbb{Z} has global dimension one. • The first
Weyl algebra A1 is a noncommutative Noetherian
domain of global dimension one. • If a ring is right Noetherian, then the right global dimension is the same as the weak global dimension, and is at most the left global dimension. In particular if a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same. • The
triangular matrix ring \begin{bmatrix}\mathbb Z&\mathbb Q \\0&\mathbb Q \end{bmatrix} has right global dimension 1, weak global dimension 1, but left global dimension 2. It is right Noetherian but not left Noetherian. == Alternative characterizations ==