Remark 2

If M is a finitely generated module over a noetherian ring A, $\dim_A(M)$ is the least upper bound of the numbers $\dim(A/p)$, where p ranges over the set $\operatorname{Ass}_A(M)$ of prime ideals of A associated with M (IV, § 1, n° 4, th. 2).

1 If A is a field, the Krull dimension of M is $\leq 0$. Care must be taken not to confuse the Krull dimension of M with the dimension (or rank) of the vector space M over the field A (A, II, p. 97, déf. 1).