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Let $M$ be a module over a commutative ring with unity. If $M\subset Q$, where $Q$ is an injective module, we know that $Q$ contains an injective hull of $M$. My question is: Can $Q$ contain several injective hulls of $M$? Put in another way, if $E$ is an injective hull of $M$, we know that any homomorphism from $M$ to an injective module extends to $E$. Is such an extension unique?

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It is mildly confusing to state your question in two different ways such that an answer of "no" to each question means different things. –  Qiaochu Yuan Jun 8 '11 at 16:51
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up vote 3 down vote accepted

No.

It would follow that the only endomorphism of $E$ which is the identity on $M$ is the identity. But the injective hull of $M=\mathbb Z/2\mathbb Z$ is the Prüfer $2$-group $C_{2^\infty}$, the $2$-primary part of the abelian group $\mathbb Q/\mathbb Z$; see Lam, Lectures on modules and rings, Example 3.36. Now multiplication by $-1$ is a non-trivial endomorphism of $C_{2^\infty}$ which is trivial on $M$.

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