Cellular covers of cotorsion-free modules. (Q2891274)

The investigations of the work of \textit{R. Göbel}, [Forum Math. 24, No. 2, 317-337 (2012; Zbl 1281.20065)] are continued.NEWLINENEWLINE We note the following three results:NEWLINENEWLINE Theorem 3.2. Let \(R\) be a commutative, torsion-free, reduced ring with 1, of size \(<2^{\aleph_0}\). Let \(K\) be any torsion free and reduced \(R\)-module of rank \(k<2^{\aleph_0}\). Then there is a cotorsion-free \(R\)-module \(G\) of rank 3 if \(k=1\), and of rank \(3k+1\) if \(2\leq k<2^{\aleph_0}\), with submodule \(K\) such that \(\Hom(G,K)=0\) and \(\Hom(G,G/K)=\pi R\) where \(\pi\colon G\to G/K\) (\(g\mapsto g+K\)) is the canonical epimorphism. In particular, \(0\to K\to G\to G/K\to 0\) is a cellular exact sequence.NEWLINENEWLINE Theorem 4.1. Let \(k'<2^{\aleph_0}\) be a cardinal, \(R\) a commutative, torsion-free ring with \(1\neq 0\), of size \(<2^{\aleph_0}\). Moreover, let \(H\) be a cotorsion-free \(R\)-module which is \(k'\)-generated such that \(\text{End}(H)=R\). Then for any cardinal \(k\) with \(k'\leq k<2^{\aleph_0}\) there is a cotorsion-free \(R\)-module \(G\) with the following properties: (i) \(G\) is \(k\)-generated and there is \(K\subseteq G\) with \(G/K=H\). (ii) If \(\varphi\in\text{End}(G)\), then there is a unique element \(r\in R\) such that \((\varphi-r\cdot id_G)(K)=0\), so there is an induced homomorphism \(\varphi_r\colon H\to G\) (\((g+k)\mapsto(\varphi-r)g\)). (iii) If \(\Hom(H,G)=0\), then \(\text{End}(G)=R\). (iv) \(\Hom(G,H)=\pi R\) where \(\pi\colon G\to H\) (\(g\mapsto g+K\)) is the canonical epimorphism.NEWLINENEWLINE In particular, if \(\Hom(H,G)=0\), then \(0\to K\to G\to H\to 0\) is a cellular exact sequence.NEWLINENEWLINE Theorem 4.4. Let \(R\) be a commutative, torsion-free ring with \(1\neq 0\) and \(k\) a cardinal with \(k^{\aleph_0}>|R|\). Moreover, let \(H\) be a cotorsion-free \(R\)-module such that \(\text{End}(H)=R\) and \(|H|\leq K\). Then there is a cotorsion-free \(R\)-module \(G\) of size \(|G|=K^{\aleph_0}\) with the following properties: (i) There is a submodule \(K\subseteq G\) with \(G/K=H\). (ii) If \(\varphi\in\text{End}(G)\), then there is a unique element \(r\in R\) such that \((\varphi-r\cdot id_G)(K)=0\), so there is an induced homomorphism \(\varphi_r\colon H\to G\) (\((g+K)\mapsto(\varphi-r)g\)). (iii) if \(\Hom(H,M)=0\), for every \(\aleph_0\)-free module \(M\), then \(\text{End}(G)=R\), and \(\Hom(G,K)=0\). (iv) \(\Hom(G,H)=\pi R\) where \(\pi\colon G\to H\) (\(g\mapsto g+K\)) is the canonical epimorphism.NEWLINENEWLINE In particular, if \(\Hom(H,M)=0\), for all \(\aleph_0\)-free modules \(M\) then \(0\to K\to G\to H\to 0\) is a cellular exact sequence.