Modules with flat socles and almost excellent extensions (Q1373095)

A unitary ring extension \(S\geq R\) is a finite normalizing extension in case there is a finite subset \(\{s_1,\dots,s_n\}\subseteq S\) such that \(S=\sum^n_{i=1} s_iR\) and \(s_iR=Rs_i\) for \(i=1,\dots,n\); and the ring \(S\) is right \(R\)-projective in case for any right \(S\)-module \(M_S\) with an \(S\)-submodule \(N_S\) then \(N_R|M_R\) implies that \(N_S|M_S\) where \(N|M\) means that \(N\) is a summand of \(M\). A finite normalizing extension \(S\geq R\) is an almost excellent extension [\textit{W. Xue}, Acta Math. Vietnam. 19, No. 2, 31-38 (1994)] in case \(_RS\) is flat, \(S_R\) is projective, and the ring \(S\) is right \(R\)-projective. If \(S\geq R\) is an almost excellent extension, the author proves that (1) an \(S\)-module \(M_S\) is flat (has flat socle) if and only if \(M_R\) is flat (has flat socle), and (2) \(S\) is right coherent (left semihereditary) if and only if \(R\) is right coherent (left semihereditary). These are proved independently by the reviewer [Algebra Colloq. 3, No. 2, 125-134 (1996; Zbl 0851.16024)].