I have a valuation ring $A$, a full abelian subcategory $\mathcal{C}$ of $\text{Mod}_A$ and an object $M$ in $\mathcal{C}$. I know that $M$ is not a projective $A$-module, but that it can still be a projective object in a full subcategory nonetheless.
Given that $\mathcal{C}$ is not the category of all modules over a ring, I can't see anything that can be used to check injectivity or projectivity aside from the explicit definition in terms of monomorphisms and epimorphisms. For example, Baer's Criterion applies to checking injectivity in $\text{Mod}_A$, but not $\mathcal{C}$.
The precise details of the case I am looking at are as follows:
Let $K$ be the field of Hahn series in an indeterminate $t$, with exponents in $\mathbb{R}$, coefficients in $\mathbb{F}_2$ and where we denote the valuation by $\nu$. Define $$A:=\{a\in K:\nu(a)\geq 0\},$$ $I_q:=t^qA$ and $I_{>q}:=\bigcup_{r>q}I_r$.
Now let $\mathcal{C}$ be the strictly full additive subcategory generated by $A/I_q$, $A/I_{>q}$, $I_{>0}/I_q$, $I_{>0}/I_{>q}$, where $0<q\leq 1$ (but we also include $A/I_{>0}$). I am trying to determine whether or not $I_{>0}/I_{>1}$ is projective and/or injective in $\mathcal{C}$.
Any input much appreciated!