I'll follow up on what Karl said with an example closer to my own experience. Let Z be the ring of integers and p a positive prime. Then Z/pZ is injective as a Z/pZ - module, being a vector space over a field, whence Z/pZ is its own injective envelope (hull) as a Z/pZ module. However, the injective envelope of Z/pZ as an abelian group is Z(p^{infty}), which gives witness to Karl's statement that the injective envelope over A can be much larger than the injective envelope over A/ann(M). You can play this game with A any commutative Noetherian ring with 1, ann(M) = any maximal ideal of R, and M = A/I where I is the chosen maximal ideal. Karl's example presents very limited choice for I since k[[x]] is local. I think Proposition 2.27 and Lemma 4.24 of "Injective Modules" by Sharpe and Vamos present enough to figure out what is going on in the general case.

I'll follow up on what Karl said with an example closer to my own experience. Let $\mathbb{Z}$ be the ring of integers and $p$ a positive prime. Then $\mathbb{Z}/p\mathbb{Z}$ is injective as a $\mathbb{Z}/p\mathbb{Z}$ - module, being a vector space over a field, whence $\mathbb{Z}/p\mathbb{Z}$ is its own injective envelope (hull) as a $\mathbb{Z}/p\mathbb{Z}$ module. However, the injective envelope of $\mathbb{Z}/p\mathbb{Z}$ as an abelian group is $\mathbb{Z}(p^{\infty})$, which gives witness to Karl's statement that the injective envelope over $A$ can be much larger than the injective envelope over $A/\mathrm{ann}(M)$. You can play this game with $A$ any commutative Noetherian ring with 1, ann($M$) = any maximal ideal of $R$, and $M = A/I$ where $I$ is the chosen maximal ideal. Karl's example presents very limited choice for $I$ since $k[[x]]$is local. I think Proposition 2.27 and Lemma 4.24 of "Injective Modules" by Sharpe and Vamos present enough to figure out what is going on in the general case.