Suppose $A$ is PID, $I\subset A$ is a nonzero ideal, show $A/I$ is an injective $A/I$-module.
I tried to prove this, but got stuck in my following argument.
To show $A/I$ is injective as $A/I$-module, by the equivalence definition of injective module, we only need to show that if $M$ is an $A/I$-module such that $A/I\subset M$, then $M$ decomposes as $M=A/I\oplus N$ for some submodule $N$.
First, $M$ can be considered as an $A$-module, and suppose $M$ is finitly generated,
then by the structure theorem of f.g. modules over PID, $M=A^n\oplus A/J_1\oplus \cdots\oplus A/J_k$, then since $M$ is an $A/I$-module, then we can assume $n=0, I\subset J_i, \forall 1\leq i\leq k$, and we are left to use $A/I\subset M$.
I think we can say there must some $J_i=I$, but how to deduce this?
For example, consider $k=1$, then it says $A/I\subset A/J, I\subset J$ implies $I=J$.