I am trying to prove or disprove that $\Bbb Z_n $ is an injective $\Bbb Z_n$-module, where $n$ is an integer $>1$.
If $n$ is a prime, then $\Bbb Z_n$ is a field, so every $\Bbb Z_n$-module is free, and hence projective. I know that for a ring $R$, every $R$-module is projective iff every $R$-module is injective, so in this case $\Bbb Z_n$ is an injective $\Bbb Z_n$-module.
But I have no idea to handle the case where $n$ is not a prime. Any hints?
Best Answer
$\newcommand\ideal{\mathfrak}$This holds, in general, on every PID $A$.
Let $\ideal a$ be an ideal of $A/Aa$ and $\varepsilon:\ideal a\to A/Aa$ be an $A/Aa$-module homomorphism. Then $\ideal a=Ab/Aa$ for some $b\in A$ and $a\in Ab$, so that $a=bc$ for some $c\in A$. Let $y\in A$ such that $\varepsilon(b+Aa)=y+Aa$. Then there exists $d\in A$ such that $cy=da=dbc$ that's $y=bd$. Thus the $A$-module homomorphism \begin{align} &A\to A/Aa&&x\in A\mapsto dx+Aa \end{align} factors through a projection $A\to A/Aa$ modulo $Aa$ giving rise to an $A$-module homomorphism $A/Aa\to A/Aa$ extending $\varepsilon$.