Selfinjectivity of rings relative to Lambek torsion theory (Q1375766)

A ring \(R\) is called left \(\tau\)-selfinjective if \(\text{Ext}^1_R(X,R)\) is torsion for every left \(R\)-module \(X\), where the prefix ``\(\tau\)-'' means ``relative to Lambek torsion theory''. The main aim of the paper is to characterize left \(\tau\)-selfinjective rings by a certain kind of linear compactness. It is known that \(R\) is left selfinjective if and only if \(R\) is left absolutely pure and right semicompact. But this is not true for the relative case. So the author needs a rather strong notion of linear compactness to characterize left \(\tau\)-selfinjective rings. He shows that \(R\) is left \(\tau\)-selfinjective if and only if \(R\) is \(\tau\)-absolutely pure and \(\varprojlim\pi_\lambda\) is a \(\tau\)-epimorphism for every inverse system of \(\tau\)-epimorphisms \(\{\pi_\lambda\colon F_\lambda\to M_\lambda\}_{\lambda\in\Lambda}\) in \(\text{Mod} R^{op}\) with the \(F_\lambda\) finitely generated free and the \(M_\lambda\) torsionless. The author is also concerned with an arbitrary class \(\mathcal C\) of left \(R\)-modules which contains \(_RR\) and is closed under taking factor modules and extensions and asks when every submodule \(X\) of \(E(_RR)\) with \(X\in{\mathcal C}\) is torsionless. In particular, he studies the class of all \(\tau\)-finitely generated modules.