Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = R$. Notice that in $K$-theory one usually considers the associated ring of this semiring (or related semirings such as the semiring of finitely generated projective modules). This, however, loses a lot of information.
I would like to know what is known about $G(R)$, where it is studied in the literature and, above all, if there is anything known about the solutions of polynomial equations (such as $X=X^2+1$ or $XY=Y^2+1$) in this semiring.
Here is an example: Let $M$ be a finitely generated $R$-module such that $M^2=M$ in $G(R)$. i.e. $M \otimes M \cong M$ in $\mathsf{Mod}(R)$. Then $M$ is cyclic: We may assume $\mathrm{Ann}(M)=0$ and hence $\mathrm{supp}(M)=\mathrm{Spec}(R)$. For $\mathfrak{p} \in \mathrm{Spec}(R)$ we have $\dim_{\kappa(\mathfrak{p})}(M \otimes_R \kappa(\mathfrak{p}))=1$, so that by Nakayama $M_\mathfrak{p}$ is generated by a single element. Since $\mathrm{Ann}(M_{\mathfrak{p}})=0$, it follows that $M_\mathfrak{p}$ is free of rank $1$. Hence, $M$ is locally free of rank $1$, hence invertible. But then $M \otimes M \cong M$ implies $M \cong R$. Conversely, every cyclic $R$-module $M$ satisfies $M \otimes M \cong M$ because $R/I \otimes R/I \cong R/I$.
Another example is the equation $X^2=0$, which only has the trivial solution $X=0$ (again by Nakayama).
Since cyclic modules and free modules are a bit boring, let me specify that I would like to know examples of equations whose solutions are interesting. For example, are flat modules definable? This would be very surprising.