Localization of A Module - Remarks

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.

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