# nonzero torsion elements

If $R$ has zero divisors show that every nonzero $R-$module has nonzero torsion elements.
If $R$ has zero divisors show that every nonzero $R-$module has nonzero torsion elements.
so there exist nonzero elements $r,s \in R$ such that $rs=0.$ let $M \neq (0)$ be an R-module and $0 \neq m \in M.$ if $sm=0,$ then we're done. so suppose $m'=sm \neq 0.$ then: $rm'=(rs)m=0. \ \ \Box$