# Non-linear Additive Maps

While teaching an introductory linear algebra course, a colleague noticed that most of the examples of additive maps he gave turned out to be linear. He asked whether I could think of a map which was additive, but not linear. In a general context, the question was to find a ring $$R$$, $$R$$-modules $$V$$ and $$W$$, and a map $$f \colon V \to W$$ such that $$f$$ is a group homomorphism, but not an $$R$$-module morphism, i.e, $$f(x+y)=f(x)+f(y)$$ for all $$x,y \in V,$$ but there is some $$r \in R$$ and $$z \in V$$ such that $$f(r \cdot z) \neq r \cdot f(z)$$.

One example that came to mind was viewing $$\mathbb{C}$$ as a vector space over itself and taking $$f \colon \mathbb C \to \mathbb C$$ to be the reflection $$f(a+bi):=b+ai$$. This map is readily seen to be a group endomorphism of $$(\mathbb C, +)$$, but it does not commute with rotation counter-clockwise by by $$\pi/2$$ radians, which is just multiplication by $$i$$. In particular, $$f(i \cdot 1)=f(i)=1 \neq -1 = i^2 = i \cdot f(1)$$.