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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.

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    $\begingroup$ Your best guess is right. A zero-divisor is a torsion element in the ring itself. $\endgroup$ – darij grinberg Jul 29 '15 at 2:18
  • $\begingroup$ I get the feeling you didn't read the definitions very carefully: how could they be considered the same? They're of a similar flavor, of course... but the definitions aren't identical... $\endgroup$ – rschwieb Jul 30 '15 at 17:25
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"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."

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