Gelfand–Kirillov dimension

Summary

The Gelfand–Kirillov dimension of an algebra over a field is the exponent of the polynomial rate at which the algebra grows. It is defined through the growth of finite-dimensional subspaces under multiplication, so it makes sense for noncommutative algebras. For a finitely generated commutative algebra it equals the Krull dimension, and the $n$-th Weyl algebra has Gelfand–Kirillov dimension $2n$. It is a basic invariant in noncommutative ring theory and in the theory of $D$-modules.