A generalization of the Dade's theorem on localization of injective modules (Q2711339)

In general the localization does not preserve the injectivity. For commutative rings, \textit{E. C. Dade} [J. Algebra 69, 416-425 (1981; Zbl 0461.13003)] has given a necessary and sufficient condition which assures the preservation of the injectivity. In this paper, let \(A\) be a ring, \(\mathcal F\) a class of right ideals of \(A\) which defines a hereditary torsion theory on the category of right \(A\)-modules. For a right ideal \(I\), fix a representation of \(I\): NEWLINE\[NEWLINE0\longrightarrow K\longrightarrow F\longrightarrow I\longrightarrow 0,\tag{*}NEWLINE\]NEWLINEwhere \(F\) is a projective module. Consider the following condition:NEWLINENEWLINELet \(L\) be a submodule of \(K\) such that \(K/L\) is \(\mathcal F\)-torsion. Then, there exists \(B\) in \(\mathcal F\) containing \(I\) and if we construct the row exact commutative diagram: NEWLINE\[NEWLINE\begin{tikzcd} NEWLINE0\ar[r] & H\ar[r,"\alpha_2"] & G\ar[r,"\beta_2"]&B\ar[r]&0\\NEWLINE0\ar[r] &K\ar[u,"k"] \ar[r,"\alpha_1" '] & F\ar[u,"j"] \ar[r,"\beta_1" ']&I\ar[u,"i"] \ar[r]&0 ,NEWLINE\end{tikzcd}\tag{**}NEWLINE\]NEWLINEwhere \(G\) is projective and \(i\) is the inclusion, there exist submodules \(M\) and \(N\) of \(F\) and \(H\) respectively such that \(F/M\) and \(H/N\) are \(\mathcal F\)-torsion and \(k^{-1}(N)\cap\alpha^{-1}_1(M)\subseteq L\).NEWLINENEWLINEThe authors show after Dade that the localization \(E_{\mathcal F}\) of any injective module \(E\) is an injective \(A_{\mathcal F}\)-module if and only if each right ideal \(I\) of \(A\) has a presentation (*) satisfying (**). In that case any presentation (*) of any such right ideal satisfies (**).