A version of purity on local abelian groups (Q2039352)

All groups in this paper are \(p\)-local abelian group (where \(p\) is a fixed prime). In [\textit{P. Keef}, J. Pure Appl. Algebra 144, No. 3, 255--276 (1999; Zbl 0941.20064)] the author defined using functorial methods some generalizations of purity. The main purpose of the paper is to study different examples of these notions of purity. He researches \(p_W^{<\lambda}\)-purity, \(L_{\lambda}\)-purity. It is proved that the reduced group \(A\) is balanced projective if and only if \(A/p^{\lambda} A\) is \(p_W^{<\lambda}\)-pure projective for every limit ordinal \(\lambda\). \(p_W^{<\lambda}\)-purity is hereditary if and only if \(\lambda\) has countable cofinality. \(p_W^{<\lambda}\)-purity is strongly hereditary if and only if \(\lambda=\omega\). It is described the Warfield groups with the \(L_{\lambda}=p_W^{<\lambda}\)-purity property.