Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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 $k[X]/(X^n)$.

share|improve this answer
    
Thanks! I see your answer can be made even simpler by taking $(A,\mathfrak{m})=(k,0)$. –  ashpool Oct 13 '11 at 16:13
    
@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 6:49
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.