On Lambek torsion theories
Let \(R\) be a ring with identity, and let \(\varepsilon_ X: X\to X^{**}\) be the canonical evaluation map from \(X\) to its double dual. Recall that the Lambek torsion theory \(\tau\) is defined via \(\text{Hom}_ R(\tau(X),E(R)) = 0\) and \(E(R)\) cogenerates \(X/\tau(X)\). The following statements are equivalent: (1) \(\tau(X) = \text{ker }\varepsilon_ X\) for every finitely presented \(X \in R\)-mod; (2) \(\tau(M) = \text{ker }\varepsilon_ M\) for every finitely presented \(M\in \text{mod-}R\); (3) \(f^{**}\) is monic for each \(f: X\to Y\) with \(_ RX\) finitely generated and \(_ RY\) finitely presented; (4) \(g^{**}\) is monic for each \(g: M\to N\) with \(M_ R\) finitely generated and \(N_ R\) finitely presented. If \(R\) is right coherent, then the following statements are equivalent: (1) \(E(_ RR)\) is flat; (2) there is a faithful, injective, flat left \(R\)-module; (3) \(\tau(X)=\text{ker }\varepsilon_ X\) for every finitely presented \(X\in R\)-mod. If \(R\) is right noetherian, then \(E(_ RR)\) is flat if and only if every finitely generated submodule of \(E(R_ R)\) is torsionless. If \(R\) is left and right coherent, then the following statements are equivalent: (1) weak \(\dim E(_ RR) \leq 1\); (2) \(\varepsilon_ X\) is an essential monomorphism for every torsionless \(X\in R\)-mod; (3) \(\text{Ext}^ 1_ R(-,R)^*\) vanishes on \(R\)-mod. The vanishing of \(\text{Ext}^ 1_ R(- ,R)^*\) on \(R\)-mod is not symmetrical. The author also notes that S. Takashima has been able to weaken the overall hypotheses in the second and third results stated in this review.
faithful, injective, flat left \(R\)-module
scientific article; zbMATH DE number 219886