Definition 2.8.29.1.2. Let
R be a
ring. We define:
LMod(R) :=: {[M, ⊕, 0, ⊖, ⊙]R | M is a set, ⊕ : M × M → M is an operation on M, 0 ∈ M, ⊖ : M → M is a function, ⊙ : (R) × M → M is an operation such that (M, ⊕, 0, ⊖, ⊙) forms a left R-module} [M, ⊕, 0, ⊖, ⊙]R = [N, ⋆, 0, ∼, ∗]R :⇔ ∃ φ : M ↔ N : [⊕ ≃φ ⋆ and 0 ≃φ 0 and ⊖ ≃φ ∼ and ⊙ ≃φ ∗] (M is a set, ⊕ : M × M → M is an operation on M, 0 ∈ M, ⊖ : M → M is a function, ⊙ : (R) × M → M is an operation, N is a set, ⋆ : N × N → N is an operation on N, 0 ∈ N, ∼ : N → N is a function, ∗ : (R) × N → N is an operation with suitable conditions) We write “let M be a left R-module” for “let M ∈ LMod(R).”
References.