A module over a unit ring R is called divisible if, for all r element R which are not zero divisors, every element m of M can be "divided" by r, in the sense that there is an element m' in M such that m = r m'. This condition can be reformulated by saying that the multiplication by r defines a surjective map from M to M. It can be shown that every injective R-module is divisible, but the converse only holds for particular classes of rings, e.g., for principal ideal domains. Since Q and Q/Z are evidently divisible Z-modules, this allows us to conclude that they are also injective. An additive Abelian group is called divisible if it is so as a Z-module.
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