Find an example of a commutatice unital ring $R$ and an $R$-module such that $\text{Ass}(M)=\emptyset$

Question

Find an example of a unital commutative ring $R$ and an $R$-module $M$ such that $\text{Ass}(M)=\emptyset$.

Recall that

• $\text{Ass}(M)=\{p\in\text{spec}(R):\exists 0\ne m\in M:\text{ann}(m)=p\}$.
• $\text{ann}(m)=\{r\in R:rm=0\},\forall m\in M$.

I thought of $R=\mathbb Z$ and $M=\mathbb Z/n\mathbb Z$, which doesn't work.

Answer 1

Take $A_n=K[[X^{2^{-n}}]]$, and $A=\bigcup_n{A_n}$. Let $M=R=A/I$, where $I$ is the ideal generated by the $X^{\alpha}$, $\alpha \geq 1/3$.

Edit: note that if $m \in M$ is nonzero with annihilating ideal is $I$, and if there is no $m’ \in m$ of which the annihilating ideal contains strictly $I$, then $I \in Spec(R)$. So we have to look for not-Noetherian rings, hence the complicated example.