Rings that cannot be representations rings Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo the relations $[M\oplus V]=[M]+ [V]$, and the multiplication is given by the tensor product.
Not all rings can be realized as the representation ring of some monoidal category $\mathcal{C}$. However I cannot seem to find some nice examples of this fact in literature. I'd very much like to see an example and the reasoning why this is true. Any references are very welcome.
Thank you. 
 A: If $\mathcal{C}$ is a monoidal Krull-Schmidt category with finitely many indecomposables, then this translates to $r(\mathcal{C})$ being a free $\mathbb{Z}$-module of finite rank, so clearly an interesting example will be of this form.
Take the ring to be $\mathbb{Z}[i]$ and note that if this arises as $r(\mathcal{C})$ for some category as above then this category will have precisely two indecomposables, and one of these must be the neutral element for the tensor product in $\mathcal{C}$, which must decategorify to the unit in the ring. Let $X$ be the other indecomposable and assume that $X$ decategorifies to some $a + bi$ in $\mathbb{Z}[i]$ (we can clearly assume that $b\neq 0$).
Now we also note that in general, the $\mathbb{Z}$-basis of $r(\mathcal{C})$ given by the decategorifications of the indecomposables will be positive, since clearly there is no such thing as minus in the category. But it is a basic exercise in linear algebra to check that $(a+bi)^2 = (-a^2 - b^2)\cdot 1 + 2a\cdot (a + bi)$ so the structure constants will never be non-negative no matter which element we try. Thus, $\mathbb{Z}[i]$ cannot be the representation ring of such a category.
The existence of a positive basis as mentioned above is one of the very strong features of categorification, though there are still plenty of things that can happen for positive based algebras that do not correspond to anything coming from the categorical picture (though the examples I am mainly familiar with are ones where the algebra is indeed the decategorication of some $2$-category but where some representation of the algebra cannot come from a $2$-representation of the $2$-category).
