The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $C$. Recall that the uniform dimension of $c \in C$ is the supremum of the $n$ such that there exists a monomorphism $c_1 \oplus \cdots \oplus c_n \to c$. Equivalently it’s the number of factors in a finest direct sum decomposition of the injective hull of $c$.

The uniform dimension is subadditive in short exact sequences and monotonic in subobjects but not necessarily in quotients. That is, if $d$ is a subobject of $c$ then the dimension of $c$ is at least the dimension of $d$, at most the sum of the dimensions of $d$and of $c/d$, but not necessarily at least the dimension of $c/d$. For example the dimension of the integers as an abelian group is 1 but the dimension of the integers mod 6 is 2.

However, this pathological example seems attributable to the fact that the category of abelian groups has more than one indecomposable injective. In the category of $p$-local abelian groups for a prime $p$ it seems that maybe we do have that the dimension of $c$ is at least the dimension of any quotient? More generally, I wonder:

Question: Let $C$ be a Grothendieck category with a unique indecomposable injective. Then for $c \in C$ and $c/d$ a quotient, is $dim(c) \geq dim(c/d)$?

I’m happy to understand special cases like assuming $C$ is a category of modules, or alternatively assuming $C$ is locally coherent or something.