Let $M$ be an $A$-module. Is its injective hull affected by whether I regard $M$ as an $A$-module or $A/\mbox{Ann}(M)$-module ?

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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.

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Yes, take $A = k[[x]]$ and $M = A/(x)$. Then as a $k = A/(x) = A/\text{Ann}(M)$-module, the injective hull of $k$ is $k$. As an $A$-module, the injective hull is much much bigger.

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whate\ injective hull of field K is K?

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