Pure-periodic modules and a structure of pure-projective resolutions.
All the modules are right \(R\)-modules over an associative ring \(R\) with identity. Recall that an exact sequence \(\cdots\to X_{n-1}\to X_n\to X_{n+1}\to\cdots\) is said to be pure if the induced sequence \(\cdots\to X_{n-1}\otimes_RL\to X_n\otimes_RL\to X_{n+1}\otimes_RL\to\cdots\) of Abelian groups is exact for any left \(R\)-module \(L\). An epimorphism \(f\colon Y\to Z\) is said to be pure if the exact sequence \(0\to\text{Ker\,}f\to Y\to Z\to 0\) is pure. A module \(P\) is said to be pure-projective if for any pure epimorphism \(f\colon Y\to Z\) the induced group homomorphism \(\Hom_R(P,f)\colon\Hom_R(P,Y)\to\Hom_R(P,Z)\) is surjective. The main results of the paper are the following ones. Let \(M\) be a right module and let \(\cdots\to P_n@>{d_n}>>P_{n-1}@>{d_{n-1}}>>\cdots@>{d_2}>>P_1@>{d_1}>> P_0@>{d_0}>>M\to 0\) be a pure exact sequence such that the modules \(P_0,\dots,P_n,\dots\) are pure-projective. Then, for each \(n\geq 0\), the \(n\)-th pure syzygy module \(\text{Ker\,}d_n\) of \(M\) is an \(\aleph_n\)-directed union of \(\aleph_n\)-generated pure-projective \(R\)-modules, which are pure submodules of \(P_n\) and of \(\text{Ker\,}d_n\) (Theorem 1.2). Let \(M\) be a right \(R\)-module. If there exists a pure-exact sequence \(0\to M\to P_n\to P_{n-1}\to\cdots\to P_1\to P_0\to M\to 0\) of modules such that the modules \(P_0,\dots,P_n\) are pure-projective, then \(M\) is pure-projective (Theorem 1.3).
scientific article; zbMATH DE number 2101021