Let $(A,\mathfrak{m})$ be a noetherian local ring, and $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. I'm pretty sure that $E(A/\mathfrak{m})$ doesn't automatically extend to an $A/\mathfrak{m}$-module via the projection map $A\twoheadrightarrow A/\mathfrak{m}$. That is, the elements of $\mathfrak{m}$ do not necessarily kill $E(A/\mathfrak{m})$. Can anyone think of an easy example of this?
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Take $k[X]/(X^n)$. This is a noetherian local ring. The injective hull of $k\cong (k[X]/(X^n))/(X)$ is $k[X]/(X^n)$, but $X$ does not kill $k[X]/(X^n)$.
Julian Kuelshammer
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Thanks! I see your answer can be made even simpler by taking $(A,\mathfrak{m})=(k,0)$. – ashpool Oct 13 '11 at 16:13
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@ashpool I don't think so, the injectie hull of $k$ is $k$ and this is a module for $k/0\cong k$, the projection map is an isomorphism in your case. – Julian Kuelshammer Oct 14 '11 at 06:49