FTF rings and Frobenius extensions (Q1604373)

Let \(R\) be a ring with identity element. Then \(R\) is called a left FTF ring if there exists a hereditary torsion theory \(\gamma\) on \(R\)-mod such that a left \(R\)-module \(M\) is \(\gamma\)-torsionfree if and only if \(M\) embeds in a flat left \(R\)-module. Let \(\lambda\) denote the Lambek torsion theory. If \(R\) is left FTF, then \(\gamma\leq\lambda\). If \(P/\lambda(P)\) is torsionless for every finitely presented \(P\in R\)-mod, then \(R\) is called almost absolutely pure. If \(R\) is \(\lambda\)-coherent, then \(R\) is called left almost coherent. The following conditions are equivalent: (1) \(R\) is FTF; (2) \(R\) is almost absolutely pure and almost coherent; (3) direct products of \(E(_RR)\) [and \(E(R_R)\)] are flat left [right] \(R\)-modules; (4) \(R\) is almost coherent and \(E(_RR)\) is flat. If \(S\) is a ring extension of \(R\) such that \(_RS\) and \(S_R\) are flat and if \(S\) is a left FTF ring, then \(R\) is also a left FTF ring. If \(R\) is a left FTF ring and if \(R\to S\) is a ring homomorphism such that \(_RS\) and \(\Hom_R({_RS},R)\) are flat left \(R\)-modules, then \(S\) is a left FTF ring. A ring extension \(R\subseteq S\) is said to be left quasi-Frobenius if \(_RS\) is finitely generated and projective and \(_SS_R\) is isomorphic to a direct summand of a direct sum of finitely many copies of \(_S\Hom_R({_RS},R)\). If \(R\subseteq S\) is a quasi-Frobenius extension, then \(S\) is almost coherent with left (or right) flat-dominant dimension \(\geq 1\) if and only if \(R\) is almost coherent with left (or right) flat-dominant dimension \(\geq 1\). If \(R\subseteq S\) is a quasi-Frobenius extension, then \(S\) has a semi-primary two-sided maximal quotient ring if and only if \(R\) has a semi-primary two-sided maximal quotient ring. If \(R\subseteq S\) is a quasi-Frobenius extension and \(R\) has a QF two-sided maximal quotient ring, so does \(S\).