Can there exist an element $r$ in the base ring $R$ such that $r.m =0$ for all $m$ in a module $M$ over $R$?

Question

Let $M$ be a nontrivial $R$ module. Then can there exist an element $r \in R$ such that $r.m=0$ for all $m \in M$?

The set of such elements is called the annihilator of $M$. You can see some examples in the previous link. — azif00, Oct 04 '22 at 02:08

score 4 · Accepted Answer · answered Oct 04 '22 at 17:58

You mean an element like $0\in R$? That happens for every module.

In fact, it is rather special for $0$ to be the only element that does that (in which case the module is called faithful.) In general the set of such elements of $R$ is called the annihilator of $M$ and it can be nonzero.

For example, with $R=F_2\times F_2$ and $M=\{0\}\times F_2$, the elements of $F_2\times\{0\}$ do that.

Can there exist an element $r$ in the base ring $R$ such that $r.m =0$ for all $m$ in a module $M$ over $R$?

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