I have a discrete valuation ring $A$ with field of fractions $K$, and a full abelian subcategory $\mathcal{C}$ of $\text{Mod}_A$ that is closed under taking submodules and quotients.
Let $M$ be an object in $\mathcal{C}$ and let $\iota$ be the map $$\iota:m\mapsto 1\otimes m:M\to K\otimes M.$$ I have shown that $\iota(M)$ is a projective object in $\mathcal{C}$.
My question is this: How can I show that $M=M_0\oplus M_1$ for some objects $M_0$, $M_1$ in $\mathcal{C}$ with $K\otimes M_1=0$?
Since $\iota(M)$ is projective in $\mathcal{C}$, there exists a morphism $\alpha:\iota(M)\to M$ for which $\iota\circ\alpha$ is the inclusion $\iota(M)\hookrightarrow K\otimes M$, so $\alpha$ must be injective. So we have two submodules $\alpha(\iota(M))$ and $\text{ker}(\iota)$. So maybe $M=\alpha(\iota(M))\oplus\text{ker}(\iota)$?
If $m\in\alpha(M_0)\cap\text{ker}(\iota)$, then $m=\alpha(1\otimes m')$ for some $m'\in M$ and $\iota(m)=1\otimes m=0$, so $1\otimes(\alpha(1\otimes m'))=0$. But I am not sure where to go from here. Any ideas?