Krull dimension of a noncommutative ring

Summary

The Krull dimension of a module over a possibly noncommutative ring, in the sense of Gabriel and Rentschler, is an ordinal. It measures how far the lattice of submodules is from satisfying the descending chain condition. It is defined by transfinite recursion: dimension $0$ means nonzero Artinian. Dimension at most $\alpha$ means that every descending chain of submodules has, from some point on, factors of dimension less than $\alpha$. The right Krull dimension of a ring is the Krull dimension of the ring as a right module over itself. For commutative Noetherian rings it agrees with the classical Krull dimension defined by chains of prime ideals.