Rings over which flat covers of simple modules are projective. (Q2892985)

It is known that the almost-perfect (\(A\)-perfect) rings (i.e. rings such that every flat \(R\)-module is \(R\)-projective) are exactly those rings \(R\) over which flat covers of finitely generated modules are projective [\textit{B. Amini, A. Amini, M. Ershad}, Commun. Algebra 37, No. 12, 4227-4240 (2009; Zbl 1190.16002) and \textit{A. Amini, M. Ershad, H. Sharif}, Commun. Algebra 36, No. 8, 2862-2871 (2008; Zbl 1155.16002)].NEWLINENEWLINE Let \(\mathbf S\) be the class of all representatives of simple right \(R\)-modules, \(R\) a ring with 1. A ring \(R\) is defined to be right \(B\)-perfect if for every flat module \(F\) and \(S\in\mathbf S\) and homomorphisms \(f\colon R\to S\), \(h\colon F\to S\), there exists a homomorphism \(g\colon F\to R\) such that \(h=fg\). It is shown that \(R\) is right \(B\)-perfect if and only if flat covers of simple modules are projective. Several other characterizations for \(R\) to be right \(B\)-perfect are also found. It is shown by means of an example that, although the class of \(A\)-perfect rings is contained in the class of \(B\)-perfect rings, the reverse inclusion is not true.NEWLINENEWLINE It is proven that a ring \(R\) is right \(B\)-perfect if and only if \(R\) is semilocal and any indecomposable flat right \(R\)-module with a unique maximal submodule is projective. Right \(B\)-perfect rings are also characterized as those rings \(R\) for which every right ideal containing \(J(R)\) is cotorsion, or equivalently, as the semilocal rings such that \(J(R)\) is cotorsion.