Remark

(2) When a free A-module E is such that any two bases of E are equipotent, the cardinal of an arbitrary basis of E over A is also called the dimension or rank of E and denoted by $\dim_A E$ or $\dim E$.

(3) Let $A$ be a ring such that any two bases of a free $A$-module are equipotent and let $K$ be a subfield of $A$, so that $A$ can be considered as a left vector space over $K$ by restricting the scalars. Every free $A$-module $E$ can similarly be considered as a left vector space over $K$ and it then follows from § 1, no. 13, Proposition 25 that

(4) In Chapter VIII we shall see examples of rings satisfying the conclusion of Proposition 3 but not the hypothesis.