Localization of Semimodules

Abstract

In Chapter 10 we constructed semirings of fractions of certain semirings. We now subsume that construction in the more general construction of localizations of semimodules over semirings. Our method follows the method for modules over rings given in [Golan, 1986]. If R is a semiring then a nonempty subset k of lideal(R) is a topologizing filter if and only if the following conditions are satisfied:

• (1) If I ⊆ H are left ideals of R with I ∈ k then H ∈ k;

• (2) If I, H ∈ K then I ∩ H ∈ K;

• (3) If I ∈ K and a ∈ R then (I: a) ∈ K.