Let $(R,m)$ be a commutative Noetherian local ring with $\operatorname{depth}(R)>0$ and $M$ be a finitely generated $R$-module with $\operatorname{depth}(M)=0$. Then can we take an $R$-regular element $x\in m$ such that $x \notin \bigcup_{p\in \operatorname{Ass}(M)-\{m\}}p$?
Thank you
$\DeclareMathOperator{\Ass}{\operatorname{Ass}}$If $\Ass(M) = \{m\}$, then $\displaystyle \bigcup_{p \in \Ass(M) - \{m\} }p = \emptyset$, so any $R$-regular element $x \in m$ (which exists since $\text{depth}(R) > 0$) will do. On the other hand if $\{m\} \subsetneq \Ass(M)$, then $\displaystyle m \not \subset \bigcup_{p \in \Ass(M) - \{m\}}p$, so in fact $\displaystyle m \not \subset \left( \bigcup_{p \in \Ass(M) - \{m\}}p \right) \bigcup \left( \bigcup_{p \in \Ass(R)} p \right)$ (again since $\text{depth}(R) > 0$), so any element $x \in m$ outside the big union will do.