\(\Sigma\)-injective modules over left duo and left distributive rings (Q1815302)

Let \(R\) be an associative ring with 1, \(M\) a left unitary \(R\)-module. \(M\) is called \(\Sigma\)-injective if \(M^{(A)}\) is injective for any index set \(A\). \(R\) is called left duo if every left ideal of \(R\) is an ideal. \(R\) is called left distributive if the lattice of left ideals of \(R\) is distributive. The authors use the technique of noncommutative classical localizations for distributive rings and show that over any left distributive or left duo ring \(R\) there exists a \(1\)-\(1\) correspondence between \(\Sigma\)-injective indecomposable left modules and completely prime ideals \(P\) such that the left classical localization \(R_{(P)}\) exists and is a left noetherian ring. It is noteworthy that the well-known theorems of Krause and Matlis can be reformulated in terms of such a correspondence. Besides this the authors essentially clarify the structure of arbitrary \(\Sigma\)-injective left modules over any left distributive or left duo ring. In particular, the authors completely describe \(\Sigma\)-injective left modules over a left uniserial ring.