According to the definition in [Lang] p. 129: "A subset S of a module M is said to be <b>linearly independent</b> (over [the ring] A) if whenever we have a linear combination ∑<sub>x ∈S</sub> a<sub>x</sub>x which is equal to 0, then a<sub>x</sub>=0 for all x∈S." This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems.<br> <br> Usually, the linear independence is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "In the theory of vector spaces, a set of vectors is said to be <b>linearly dependent</b> if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be <b>linearly independent</b>." Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {v<sub>i</sub> | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {a<sub>j</sub> | j∈J} of elements of K, all nonzero, such that ∑<sub>j∈J</sub> a<sub>j</sub>v<sub>j</sub>=0. A set X of elements of V is linearly independent if the corresponding family{x}<sub>x∈X</sub> is linearly independent". <br> <b>Remark 1:</b> There are already definitions of (linearly) independent families (df-lindf) and (linearly) independent sets (df-linds). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds and df-lininds for (linear) independence for (left) modules is shown in lindslininds. <br> <b>Remark 2:</b> Subsets of the base set of a (left) module are <b>linearly dependent</b> if they are not linearly independent (see df-lindeps) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent to our definition for (left) vector spaces (see isldepslvec2) and not for (left) modules in general.