Remarks
- The definition applies in particular to M=R, and we get back the localized ring S−1R.
- There is a module homomorphism
-
- φ: M → S−1M
- mapping
- φ(m) = m / 1.
- Here φ need not be injective, in general, because there may be significant torsion. The additional u showing up in the definition of the above equivalence relation can not be dropped (otherwise the relation would not be transitive), unless the module is torsion-free.
- Some authors allow not necessarily multiplicatively closed sets S and define localizations in this situation, too. However, saturating such a set, i.e. adding 1 and finite products of all elements, this comes down to the above definition.
Read more about this topic: Localization Of A Module
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