Definition of a point annihilator

Question

In this paper https://www.researchgate.net/publication/312907984_On_Right_S-Noetherian_Rings_and_S-Noetherian_Modules, it states that a point annihilator of a right $R$-Module $M$ is an annihilator of a nonzero element $m$ of $M$. But I still don't understand what it means. Can anybody explain it clearly, also with an example?

Answer 1

It just means, for a given $m\in M$, $\{x\in R\mid mx=0\}$ .

You can do this for literally any element in any module. How about you pick a few simple ones.

For example, try

1. $M=R=\mathbb Z$; and
2. $R=\mathbb R\times \mathbb R$ with $M=\{0\}\times \mathbb R$ where the pairs in $R$ multiply pointwise with the pairs in $M$.