On Lambek torsion theories. III (Q1912851)

[For part II cf. ibid. 31, No. 3, 729-746 (1994; Zbl 0823.16020).] Let \(R\) be an associative ring with identity. The paper begins with a preliminary study of flat epimorphic extensions of rings and rings for which every finitely generated submodule of \(E(_RR)\) is torsionless. Let \(\tau\) denote the Lambek torsion theory. Arbitrary direct products of copies of \(E(_RR)\) are flat if and only if \(R\) is \(\tau\)-absolutely pure and right \(\tau\)-coherent. The main result of the paper is the following extension of a theorem of \textit{C. Faith} [Nagoya Math. J. 27, 179-191 (1966; Zbl 0154.03001)]: an extension ring \(Q\) of a ring \(R\) is a quasi-Frobenius maximal two-sided quotient ring of \(R\) if and only if \(R\) is left \(\tau\)-noetherian, \(_R(Q/R)\) is \(\tau\)-torsion, and \(Q_R\) is injective. The last section presents a proof of a result of \textit{K. Masaike} [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 11, 26-30 (1971; Zbl 0235.16001)].