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, \( f(i \cdot 1)=f(i)=1 \neq -1 = i^2 = i \cdot f(1)\).