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Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that

$\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ .

For what kinds of modules in general is this the case?

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A module with that property is called a "torsionfree $R$-module". Some conditions are known; e.g., a module over a Prufer domain is torsionfree if and only if it is flat. – Arturo Magidin Jul 20 '12 at 21:52
Oh, wait; you want a weaker condition than "torsion free". Essentially you are asking for modules with the condition that $ax=0$ implies $x=0$ or $a$ is a zero divisor... – Arturo Magidin Jul 20 '12 at 22:01
Thanks! But you're right, I'm looking specifically at the case where some elements of the ring may be zero divisors... – pesco Jul 20 '12 at 22:37

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