Reflexive modules over \(\text{QF-}3'\) rings (Q1916083)

Let \(R\) be a ring. For an \(R\)-module \(M\), let \({\mathcal F}_M\) denote the localizing subcategory of \(R\)-mod cogenerated by \(E(M)\). If every \(M\)-generated submodule of \(E(M)\) is \(M\)-cogenerated and \(X\in R\)-mod is an \({\mathcal F}_M\)-linearly compact module, then \(X\) is \(M\)-reflexive if and only if \(M\)-dom\(\dim X\geq 2\). If \({_RM_S}\) is a faithfully balanced bimodule such that both \(E({_RM})\) and \(E(M_S)\) are \(M\)-cogenerated and if \(X\in R\)-mod, then \(X\) is \(M\)-reflexive if and only if \(X\) is \({\mathcal F}_M\)-linearly compact and \(M\)-dom dim \(X\geq 2\). Consequently, for a QF-\(3'\) ring, \(X\) is reflexive if and only if \(X\) is Lambek linearly compact and \(R\)-dom\(\dim X\geq 2\). A maximal left quotient ring \(R\) is a QF-3 ring if and only if (1) \(R\) is left QF-\(3'\), (2) \(R\) is left Lambek linearly compact, and (3) \(R\) has a dominant right module.