(M, ⊕, 0, ⊖, ⊙) forms a left R-module :⇔ (M, ⊕, 0, ⊖) forms an abelian group and ⊙ is left-distributive over ⊕ and [∀ a, b ∈ R, x ∈ M : (a + b) ⊙ x = (a ⊙ x) ⊕ (b ⊙ x)] and [∀ x ∈ M : 0R ⊙ x = 0] and 0 is a right absorbing element for ⊙ and [∀ a ∈ R, x ∈ M : (−a) ⊙ x = a ⊙ ⊖(x) = ⊖(a ⊙ x)] and [∀ a, b ∈ R, x ∈ M : (a ⋅ b) ⊙ x = a ⊙ (b ⊙ x)] and 1R is a left identity for ⊙ ⇔ (M, ⊕, 0, ⊖) forms an abelian group and ⊙ is left-distributive over ⊕ and [∀ a, b ∈ R, x ∈ M : (a + b) ⊙ x = (a ⊙ x) ⊕ (b ⊙ x)] and [∀ a, b ∈ R, x ∈ M : (a ⋅ b) ⊙ x = a ⊙ (b ⊙ x)] and 1R is a left identity for ⊙ Equivalence. No proof.
References.