Example

Take $G =\mathbf{Z}$ and $\varphi (F) =$ dim F. The corresponding homomorphism from $\mathscr{R}(\mathfrak{a})$ to $\mathbf{Z}$ is denoted by dim. Let $c$ be the class of a trivial $\mathfrak{a}$-module of dimension 1, and let $\psi$ be the homomorphism $n \rightarrow nc$ from $\mathbf{Z}$ to $\mathscr{R}(\mathfrak{a})$. It is immediate that

dim$\circ \psi =$ Id$_{\mathbf{Z}}$,

so that $\mathscr{R}(\mathfrak{a})$ is the direct sum of Ker dim and $\mathbf{Z}c$.