$R$-linearity is equivalent to preserving expressions of the form $rx+y$.Suggest a slogan
Let \(R\) be a ring and \(M, N\) left \(R\)-modules. A map of sets \(\phi\colon M \to N\) is an \(R\)-module homomorphism if and only if \[ \phi(r x + y) = r\,\phi(x) + \phi(y) \quad \text{for all } r \in R \text{ and } x, y \in M. \]