a tensor r is zero

Question

Let $R$ be a ring and $A$ be a (right) $R$-module.

Consider $a\otimes r=0$ in $A\bigotimes_R R$.

Under what conditions (of $r$) can we conclude that $a=0_A$?

For instance if $r=1$, can we say that if $a\otimes 1=0$, then $a=0$?

Thanks for any help.

Answer 1

$A\cong A\bigotimes_R R$ as right $R$ modules via the map $a\mapsto a\otimes 1$. Thus we can conclude that when $r=1$, $a\otimes 1=0\otimes 0$ iff $a=0$.

In general, $a\otimes r=ar\otimes 1$, and this will be zero precisely when $ar=0$, i.e. when $r\in Ann(a)$, the annihilator of $a$.