A $\mathbb Z_+$-algebra is an algebra $A$ over $\mathbb C$ with given basis $\{v_i\}$ such that $$v_iv_j=\sum_k n_{ijk}v_k,\hspace{10mm}n_{ijk}\in\mathbb Z_{\geq0}.$$ An example of such an object is the character ring of a group $G$, given by $\mathbb C[\chi_\rho\ |\ \rho\in\mathrm{Irr} G]$.
A $\mathbb Z_+$-module over $A$ is a module $M$ over $A$ with given basis $\{m_i\}$ such that $$v_im_j=\sum_k n_{ijk}'m_k,\hspace{10mm}n_{ijk}'\in\mathbb Z_{\geq0}.$$ If $H\subset G$ is a subgroup, then the character ring of $H$ becomes a $\mathbb Z_+$-module over the character ring of $G$, by restriction to $H$. (so for $\rho\in\mathrm{Irr} G$ and $\sigma\in\mathrm{Irr} H$, we have $\chi_\rho\cdot\chi_\sigma=\chi_\rho\vert_{H}\cdot\chi_\sigma$)
Direct sums of $\mathbb Z_+$-modules is just the direct sum of modules, and the basis is the union of the bases. A $\mathbb Z_+$-module is said to be indecopmosable if it is not the direct sum of nonzero $\mathbb Z_+$-modules.
Now my question is this:
Are all indecomposable $\mathbb Z_+$-modules of the character ring of $G$ of the form above: so the character ring of a subgroup $H$ of $G$?
The reason I believe this is true is because a more categorial version of this is proved in EGNO's Tensor Categories, Corollary 7.12.20, but I don't understand their proof as it refers to 3 other theorems, which themselves each refer to other theorems and so on. I also think that a proof just in terms of $\mathbb Z_+$-algebras would be easier, but I don't know how to do it.