Unimodular element of a module

Question

Let $A$ be a noetherian ring and $M$ an $A$-module. An element $z \in M$ is said to be unimodular if $Az$ is a direct summand of $M$ and $Ann(z) = \{r \in A | rz=0\} = 0$. The order ideal of $M$ is: $O_M(z)=\{\phi(z)| \phi \in Hom_A(M,A)\}$.

Now the author says that an element $z \in M$ is unimodular if and only if $O_M(z)=A$. I think we need to think in terms of surjection to prove this result but I don't know exactly how should I proceed. Any help would be really appreciated.

Answer 1

There is a typo, the correct definition is “An element $z \in M$ is unimodular if $\operatorname{Ann}(z)=\{a \in A : az=0\}$ is zero and $Az$ is a direct summand of $M$”.

Here are some ideas, I hope you can connect them and finish the proof from this.

• Assuming that $\operatorname{Ann}(z)=0$, $Az$ is a direct summand of $M$ if and only if the monomorphism $A \to M$, $a \mapsto az$ splits.
• The ideal $O_M(z)$ is $A$ if and only if $1 \in O_M(z)$.