On Lambek torsion theories. II (Q1804206)

[For part I cf. the first author, ibid. 29, No. 3, 447-453 (1992; Zbl 0787.16026).] Let \(R\) be a ring with identity, and let \(\tau\) denote the Lambek torsion theory (on either \(R\)-mod or mod-\(R\)). Let \(\varepsilon_ x : X\to X^{**}\) be the usual evaluation map. Then \(R\) is called \(\tau\)- absolutely pure if any of the following equivalent conditions hold: (1) \(\tau(X) =\text{ker }\varepsilon_ x\) for every finitely presented \(X\in\text{mod-}R\), (2) every \(\tau\)-finitely presented \(X\in\text{mod- }R\) is torsionless, or (3) \(\text{Ext}_ R(X,R)\) is torsion for every finitely presented \(X\in\text{mod-}R\). If \(R\) is \(\tau\)-absolutely pure, then the following statements are equivalent: (1) \(R\) is left and right \(\tau\)-noetherian, and (2) \(R\) is left \(\tau\)-artinian. An \(R\)- homomorphism \(\pi\) is called a \(\tau\)-epimorphism if \(\text{coker }\pi\) is \(\tau\)-torsion. A module \(X\) is called \(\tau\)-semicompact if, for every inverse system of \(\tau\)-epimorphisms \(\{\pi_ \lambda : X\to Y_ \lambda\}_{\lambda\in\Lambda}\) with each \(Y_ \lambda\) torsionless, \(\varprojlim\pi_ \lambda\) is a \(\tau\)-epimorphism. The ring \(R\) is called left (right) \(\tau\)-semicompact if \(R\) is \(\tau\)-semicompact as a left (right) \(R\)-module. If \(R\) contains an idempotent \(f\) with \(RfR\) a minimal dense left ideal and \(fR\) an injective right ideal, then every \(X\in\text{mod-}R\) with \(\varepsilon_ x\) a \(\tau\)-epimorphism is \(\tau\)-semicompact (and hence every finitely generated \(X\in R\)-mod is \(\tau\)-semicompact). Also, \(R\) is \(\tau\)-absolutely pure and left \(\tau\)- semicompact if and only if \(\text{Ext}_ R (R/I,R)\) is \(\tau\)-torsion for every right ideal \(I\). When \(f\in R\) is an idempotent with \(RfR\) a minimal dense left ideal, then \(fR\) is an injective right ideal if and only if \(R\) is \(\tau\)-absolutely pure and left \(\tau\)-semicompact. A ring \(R\) is right QF-3 if and only if the following conditions hold: (1) \(R\) is \(\tau\)-absolutely pure, (2) \(R\) is left \(\tau\)-semicompact, (3) \(R\) contains an idempotent \(f\) such that \(RfR\) is a minimal dense left ideal and \(RfR\) is a semiperfect ring, and (4) every cocritical right \(R\)- module has a nonzero socle. A ring \(R\) is left and right QF-3 if and only if the following conditions hold: (1) \(R\) is \(\tau\)-absolutely pure, (2) \(R\) is left and right \(\tau\)-semicompact, and (3) \(R\) contains idempotents \(e\) and \(f\) such that \(ReR\) and \(RfR\) are minimal dense right and left ideals, respectively. If \(R\) is \(\tau\)-absolutely pure, left and right \(\tau\)-semicompact, then \(R\) has a maximal two-sided quotient ring. Other results on maximal quotient rings are also given.