Definition 3

An element x of an A-module E is said to be a linear combination with coefficients in A of a family $(a_i)_{i \in I}$ of elements of E if there exists a family $(\lambda_i)_{i \in I}$ of elements of A, of finite support, such that $x = \sum_{i \in I} \lambda_i a_i$.

In general there are several distinct families $(\lambda_i)$ satisfying this condition (cf. no. 11).

Note that 0 is the only linear combination of the empty family of elements of E (by the convention of I, § 2, no. 1).