Localization of Injective Modules over Valuation Rings

It is proved that EJ is injective if E is an injective module over a valuation ring R, for each prime ideal J 6= Z. Moreover, if E or Z is flat, then EZ is injective too. It follows that localizations of injective modules over h-local Prüfer domains are injective too. If S is a multiplicative subset of a noetherian ring R, it is well known that SE is injective for each injective R-module E. The following example shows that this result is not generally true if R is not noetherian. Example 1. Let K be a field and I an infinite set. We put R = K , J = K and S = {1− r | r ∈ J}. Then R/J ∼= SR, R is an injective module, but R/J is not injective by [5, Theorem]. However, we shall see that, for some classes of non-noetherian rings, localizations of injective modules are injective too. For instance: Proposition 2. Let R be a hereditary ring. For each multiplicative subset S of R and for every injective R-module E, SE is injective. There exist non-noetherian hereditary rings. Proof. Let F be the kernel of the natural map: E → SE. Then E/F is injective and S-torsion-free. Let s ∈ S. We have (0 : s) = Re, where e is an idempotent of R. It is easy to check that s+ e is a non-zerodivisor. So, if x ∈ E, there exists y ∈ E such that x = (s+e)y. Clearly eE ⊆ F . Hence x+F = s(y+F ). Therefore the multiplication by s in E/F is bijective, whence E/F ∼= SE. In Proposition 2 and Example 1, R is a coherent ring. By [3, Proposition 1.2] SE is fp-injective if E is a fp-injective module over a coherent ring R, but the coherence hypothesis can’t be omitted: see [3, Example p.344]. The aim of this paper is to study localizations of injective modules and fpinjective modules over a valuation ring R. Let Z be the subset of its zerodivisors. Then Z is a prime ideal. We will show the following theorem: Theorem 3. Let R be a valuation ring, denote by Z the set of zero divisors of R and let E be an injective (respectively fp-injective) module. Then: (1) For each prime ideal J 6= Z, EJ is injective (respectively fp-injective). (2) EZ is injective (respectively fp-injective) if and only if E or Z is flat. 1991 Mathematics Subject Classification. Primary 13F30, 13C11.