So I'm having some trouble seeing why the following is true immediately. So according to wikipedia;
https://en.wikipedia.org/wiki/Torsion-free_module.
The first paragraph reads:
"In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module"
I'm definitely missing something trivially obvious; but I cant see immediately why having a zero-divisor immediately leads to the conclusion that the module MUST be the zero module ?
Any help or insights is appreciated.
Cheers !
Best Answer
Suppose that $A$ has a zero divisor $a$ (so that $ba = 0$ for some nonzero $b$) and $M$ is an $A$-module. If $M \neq 0$ then let $m \in M$ be nonzero. Then it is the case that $am = 0$ or we have $am \neq 0$ and $b(am) = 0$ - in both cases something non-zero is killed by a nonzero element of $A$.