Let $C$ be a small linear category over a commutative ring $R$. A *representation* of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation $[c] = \hom_C(c,-)$. These representations are called *generators*, because every representation can be written as a quotient $\bigoplus_{i\in I} [c_i] / \bigoplus_{j\in J} [c'_j]$ for some indexing sets $I,J$ and objects $c_i,c'_j \in C$.

A representation $M$ is *finitely generated* if it can be written in such a way with $|I|<\infty$. Equivalently, $M$ is finitely generated if there is a surjection $\bigoplus_{i=1}^n [c_i] \to M$ for some finite list $\{c_i\}$ of objects in $C$. A map of representations is a surjection iff it evaluates to a surjection in $\mathrm{Mod}(R)$ when evaluated at each object in $C$.

The category $C$ is *Noetherian* if every subrepresentation of every generator is finitely generated. This is ~~the~~ **a** natural generalization of the case of algebras (which correspond to categories with only one object, hence subrepresentations are ideals).

What strategies are there to prove that some given category is Noetherian?

For example, is there a notion of "Ore extension" of categories for which Noetherian-ness is preserved?

In my case, I have a few categories that I would like to be Noetherian. They are constructed from smaller, known-to-be-Noetherian categories by some process that looks a bit like Ore extension, and then a bit not.

I'd rather not reinvent the wheel (or the stethoscope or the transistor or...) if I can help it.