I'm not understanding what the difference is between a zero divisor and a torsion element of a module. My best guess is that the torsion elements are "vectors" and zero divisors are scalars. This seems wrong to me, but I've looked over the definitions (Wikipedia) for a few days now and I can't spot the difference.
"Torsion" is more module theoretic. Most often zero divisors are talked about in the context of products of ring elements, but you can also talk about a ring element "being a zero divisor on" module elements.
The term "torsion" is much more overloaded than zero-divisor is. For example:
one definition of a torsion element of an $R$ module is that its annihilator in $R$ is an essential ideal.
or you could ask for the annihilator to contain a regular element,
or you could always define it as simply "having a nonzero annihilator."