Precovering and Preenveloping Ideals

In 8 Salce , L. ( 1979 ). Cotorsion theories for abelian groups . Symposia Mathematica 23 : 11 – 23 . Google Scholar]] L. Salce introduced the notion of a cotorsion pair (?, 𝒞) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven to be useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (?, 𝒞) of classes of R-modules is cogenerated by a set, then it is complete 1 Eklof , P. C. , Trlifaj , J. ( 2001 ). How to make Ext vanish . Bull. London Math. Soc. 33 : 31 – 41 .Crossref], Web of Science ®] , Google Scholar]]. Recently Fu, Herzog, Guil, and Torrecillas developed the ideal approximation theory 6 Herzog , I. ( 2014 ). Phantom morphisms and Salce's Lemma . Contemp. Math. 607 : 57 – 83 .Crossref] , Google Scholar]], 4 Fu X. H. , Guil Asensio , P. A. , Herzog , I. , Torrecillas , B. ( 2013 ). Ideal Approximation Theory . Adv. in Math. 244 : 750 – 790 .Crossref], Web of Science ®] , Google Scholar]]. In this article we look at a result motivated by the Eklof and Trlifaj argument for an ideal ? when it is generated by a set of homomorphisms.