definition. linear map [ca-001F]
definition. linear map [ca-001F]
Let \(R, S\) be rings, \(M\) an \(R\)-module, \(N\) an \(S\)-module. A linear map from \(M\) to \(N\) is a function \(f : M \to _{l} N\) over a ring homomorphism \(\sigma : R \to _{+*} S\), satisfying:
- \(f(x + y) = f(x) + f(y)\) for all \(x, y \in M\).
- \(f(r \bullet x) = \sigma (r) \bullet f(x)\) for all \(r \in R\), \(x \in M\).