Some precovers and preenvelopes in functor categories (Q2065358)

Let \(R\) be an associative ring with identity, \(mod-R\) be the category of finitely presented right \(R\)-modules and \((\text{mod-}R,Ab)\) be the functor category. The authors introduce the notion of \(n\)-strongly FP-injective functor as those functors isomorphic to \(-\otimes M\) with \(M\) an \(FP_n\)-injective left \(R\)-module, i.e. Ext\(_R^1(X,M)=0\) for any finitely \(n\)-presented module \(X\), and \(n\)-strongly flat functors as those isomorphic to \((-,N)\in ((mod-R)^{op},Ab)\) with \(N\) an \(FP_n\)-flat right \(R\)-module, i.e. Tor\(^R_1(X,M)=0\) for any finitely \(n\)-presented module \(X\). In Section 3, the existence of cover and envelopes relative to this class of functors is investigated. The main result establishes that both classes of functors are preenveloping and covering in their respective categories of functors. In the last section, new characterizations of \(n\)-coherent rings [\textit{D. L. Costa}, Commun. Algebra 22, No. 10, 3997--4011 (1994; Zbl 0814.13010)] and \(n\)-regular rings [\textit{Z. Zhu}, Bull. Iran. Math. Soc. 37, No. 4, 251--257 (2012; Zbl 1277.16007)] are obtained using these classes of functors.