Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$.
I was wondering if someone could help me find an example of a ring $R$ where there exists a nonzero zero-divisor $x\in R$ but $xm\neq 0$ for every $m\in M-\{0\}$ where $M$ is some nonzero $R$-module (so in some imprecise sense, $x\neq y$ for any such $y$ above).
I believe this cannot happen when $R$ is a local ring and $M$ has finite projective dimension; see Theorem 1 in these notes. So, such an example for $M$ might need to have infinite projective dimension and I was unsure where to look and how to get started.
Thanks in advance.
If $M$ does not have finite projective dimension: let $R = k[[x,y]]/(xy)$, $M = R/(y) \cong k[[x]]$. Then $x$ is a zerodivisor in $R$, but acts as a nonzerodivisor on $M$.
If $R$ is not local: let $R = k \times k$, $k$ a field, $M = k \times 0 \subseteq R$, $x = (1,0) \in R$.