Left R-Module

Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M such that ∀ r, s ∈ R and x, y ∈ M, we have:

\begin{align*} r \cdot (x + y) &= r \cdot x + r \cdot y \tag{1}\\ (r + s) \cdot x &= r \cdot x + s \cdot x \tag{2}\\ (r \cdot s) \cdot x &= r \cdot (s \cdot x) \tag{3}\\ 1 \cdot x &= x \tag{4} \end{align*}