On topological \(M\)-injective modules (Q6041581)

A topological module \(U\) over a topological ring \(R\) is called topological injective if for every topological \(R\)-module \(M\), every open submodule \(K\) of \(M\), every continuous monomorphism \(f:K\rightarrow M\) and every continuous homomorphism \(g:K\rightarrow U\), there exists a continuous homomorphism \(h:M\rightarrow U\) such that \(hf=g\). Several researchers have tried to find sufficient and necessary conditions under which an infinite direct sum of topological injective modules is also topological injective. This problem has been solved partially only for the case when the topolgy on modules is discrete. The authors of the present paper introduce a subclass of topological injective modules, called topological \(M\)-injective modules, and show that if the infinite direct sum of topological \(M\)-injective modules is an open submodule of the direct product of the same modules, then the direct sum is also a topological \(M\)-injective module.\par For a fixed topological ring \(R\) and a fixed topological \(R\)-module \(M\), a topological \(R\)-module \(U\) is called a topological \(M\)-injective module if for every open submodule \(K\) of \(M\), every continuous monomorphism \(f:K\rightarrow M\) and every continuous homomorphism \(g:K\rightarrow U\), there exists a continuous homomorphism \(h:M\rightarrow U\) such that \(hf=g\).