Linear Independence in Left Module over Domain

The articles [22], [5], [3], [2], [4], [6], [21], [16], [14], [15], [1], [17], [19], [20], [7], [8], [9], [12], [11], [10], and [13] provide the terminology and notation for this paper. For simplicity we adopt the following rules: x is arbitrary, R is an associative ring, V is a left module over R, v, v1, v2 are vectors of V , A, B are subsets of V , and l is a linear combination of A. We now define two new predicates. Let us consider R, V , A. We say that A is linearly independent if and only if: (Def.1) for every l such that ∑ l = ΘV holds support l = ∅. A is linearly dependent stands for A is not linearly independent. One can prove the following propositions: (2) If A ⊆ B and B is linearly independent, then A is linearly independent. (3) If 0R 6= 1R and A is linearly independent, then ΘV / ∈ A. (4) ∅the carrier of the carrier of V is linearly independent. (5) If 0R 6= 1R and {v1, v2} is linearly independent, then v1 6= ΘV and v2 6= ΘV . (6) If 0R 6= 1R, then {v,ΘV } is linearly dependent and {ΘV , v} is linearly dependent.