Relative coherence of rings. (Q2892986)

Relative coherence of rings, with respect to a hereditary torsion theory, has been defined before by \textit{M. F. Jones} [Commun. Algebra 10, 719-739 (1982; Zbl 0483.16027)] and by \textit{N.-Q. Ding} and \textit{J.-L. Chen} [Manuscr. Math. 81, No. 3-4, 243-262 (1993; Zbl 0802.16023)]. The definition given here differs from both of these and proves to be the most fruitful in terms of results and echoing of properties of coherence in the relative setting.NEWLINENEWLINE Let \(\tau=(\mathbf{T,F})\) be a hereditary torsion theory for the category of all left \(R\)-modules. The following concepts (defined by Jones -- see above reference) are used: A left \(R\)-module \(N\) is said to be \(\tau\)-finitely generated if \(N/N'\in\mathbf T\) for some finitely generated submodule \(N'\) of \(N\), and \(N\) is called \(\tau\)-finitely presented if there exists an exact sequence \(0\to K\to F\to N\to 0\) with \(F\) finitely generated free and \(K\) \(\tau\)-finitely generated.NEWLINENEWLINE The alternative definition for \(\tau\)-coherence reads as follows: A ring \(R\) is said to be a left \(\tau\)-coherent ring if every \(\tau\)-finitely presented left ideal is finitely presented. A left \(R\)-module \(N\) is defined to be \(\tau\)-\(f\)-injective if \(\text{Ext}_R^1(R/I,N)=0\) for any \(\tau\)-finitely presented left ideal \(I\). A right \(R\)-module \(M\) is said to be \(\tau\)-flat if \(\text{Tor}_1^R(M,R/I)=0\) for any finitely presented left ideal \(I\). It is shown that \(\tau\)-coherent rings can be characterized in numerous ways, similar to well-known characterizations of coherent rings. For example, \(R\) is shown to be left \(\tau\)-coherent if and only if any direct product of \(R_R\) is \(\tau\)-flat if and only if any direct limit of \(\tau\)-\(f\)-injective \(R\)-modules is \(\tau\)-\(f\)-injective if and only if every right \(R\)-module has an \(fp\)-flat preenvelope.NEWLINENEWLINE Some applications are also considered. Necessary and sufficient conditions relating to existence of certain \(\tau\)-flat preenvelopes or \(\tau\)-\(f\)-injective precovers are found, for a left \(\tau\)-coherent ring \(R\) to be left \(\tau\)-\(f\)-injective. The paper concludes with a proof that every \(\tau\)-finitely presented left ideal is a direct summand of \(R\) if and only if every left \(R\)-module is \(\tau\)-\(f\)-injective if and only if every right \(R\)-module is \(\tau\)-flat if and only if every nonzero right \(R\)-module contains a nonzero \(\tau\)-flat submodule.