Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is exact. My question is: if an object $P$ of $\mathcal{B}$ is projective in $\mathcal{B}$, then is it true that $P$ is projective in $\mathcal{A}$? (And what about the injective case?)


For example, consider a noetherian ring $R$, take $\mathcal{A} = \text{Mod}R$ and $\mathcal{B} = \text{mod}R$ (subcategory of finitely generated $R$-modules). Then, if $P$ is a finitely generated $R$-module which is projective in $\text{mod}R$, is it true that $P$ is a projective object in $\text{Mod}R$? (An the injective case?)


**Edit:** The statement of the second paragraph is true: use the fact that a module is projective iff it is a direct summand of a free module; for the injective case we can use Baer's criterion.



Projective (or injective) object in a subcategory
You need more assumptions for this to be true.


Consider the ring 
$$A = \begin{bmatrix} \mathbb k & \mathbb k \\ 0 & \mathbb k \end{bmatrix},$$ where $\mathbb k$ is some field, and let $\mathcal A= \operatorname{mod} A$. There is a non-split exact sequence
$$0 \to \begin{bmatrix} \mathbb k \\ 0 \end{bmatrix} \to \begin{bmatrix} \mathbb k \\ \mathbb k \end{bmatrix} \to \begin{bmatrix} \mathbb k \\ \mathbb k \end{bmatrix} / \begin{bmatrix} \mathbb k \\ 0 \end{bmatrix} \to 0$$ in $\mathcal A$,
so the simple module $M= \begin{bmatrix} \mathbb k \\ \mathbb k \end{bmatrix} / \begin{bmatrix} \mathbb k \\ 0 \end{bmatrix}$ is not projective in $\mathcal A$.


Let $\mathcal B$ be the full exact abelian subcategory of finite direct sums of $M$. Then $M$ is projective in $\mathcal B$.


(In this example $\mathcal B$ is closed under $\mathcal A$-extensions, so that condition was not so relevant.) 


