Definition 8

Let A be a ring and M a finitely generated A-module. The Krull dimension (or simply dimension $^1$) of the A-module M is the Krull dimension of the support of M (II, § 4, No. 4, Definition 5); it is denoted by $\dim_A(M)$ (or $\dim(M)$ if there is no ambiguity).

The support of the A-module A is Spec(A); the dimension of the A-module A is therefore equal to the dimension of the ring A.

Let M be a finitely generated A-module and α its annihilator; one has

(II, § 4, n° 4, prop. 17). Hence the dimension of the A-module M, the dimension of the A-module A/α, the dimension of the ring A/α and the dimension of the (A/α)-module M all coincide; it is the least upper bound of the set of lengths of chains $p_0 \subset ... \subset p_n$ of prime ideals of A such that $\alpha \subset p_0$. By prop. 6, c) of n° 3, the dimension of M is also the least upper bound of the dimensions of the rings (or of the A-modules) A/p, where p ranges over the set of prime ideals of A, minimal among those which contain α.