Rings whose cyclics have finite Goldie dimension (Q1204388)

Given two right modules \(M\) and \(N\), the authors say that \(M\) is weakly \(N\)-injective if for every homomorphism \(\varphi:N\to E(M)\) there exists a submodule \(X\subseteq E(M)\) which is isomorphic to \(M\) satisfying \(\varphi(N)\subseteq X\). And \(M\) is weakly injective if it is weakly \(N\)- injective for every finitely generated right module \(N\). The authors prove that a ring \(R\) satisfies the property that every cyclic right \(R\)- module has finite Goldie dimension if and only if every direct sum of (weakly) injective right \(R\)-modules is weakly injective; if and only if every direct sum of indecomposable injective (weakly injective) right \(R\)-modules is weakly \(R\)-injective.