On the splitting property of rings with restricted class of injectivity domains (Q6636349)

Recall that a right \(R\)-module \(M\),the class of all right \(R\)-modules relative to which \(M\) is injective, denoted by \({\mathfrak{In}}^{-1}(M)\), is called the \textit{injectivity domain} of \(M\); that is, \({\mathfrak{In}}^{-1}(M)=\{N\in \text{Mod-}R\mid M\text{ is }N\text{-injective}\}\). Also, the collection\Nof all injectivity domains of right \(R\)-modules, denoted by \(i\mathcal{P}(R)\).\N\NIt is shown that the injective profile (or simply, the profile) of the ring \(R\) may happen to be a useful structure while investigating properties of \(R\). The significance of the profile \(i\mathcal{P}(R)\) often comes from the fact that it is a coatomic sublattice of the lattice of hereditary pretorsion classes of right \(R\)-modules (see Theorem 2.9 of [\textit{S. R. López-Permouth} and \textit{J. E. Simental}, J. Algebra 362, 56--69 (2012; Zbl 1284.16003)]).\N\NThe aim of this paper is to bring all these studies to a common denominator and try to explain why this phenomenon occurs each time we consider the same type of problem. The author comes up with the idea that the profile of the ring has exactly one coatom and call such \textit{profiles local}. It is obtained a decomposition theorem for rings with local profile. Finally, the author studies indecomposable rings and explores rings with local profiles.