The opposite of projectivity by proper classes (Q6608202)

Over the last number of years, two approaches have been suggested to measure ``how projective'' an \(R\)-module is, thereby also defining ``opposites'' of projectivity. These approaches gave rise to the projectivity domain and \(p\)-poor modules (see [\textit{C. Holston} et al., J. Pure Appl. Algebra 216, No. 3, 673`--678 (2012; Zbl. 1251.16002).]) and the subprojectivity domain en \(p\)-indigent modules (see [\textit{C. Holston} et al., Glasg. Math. J. 57, No. 1, 83--99 (2015; Zbl. 1320.16002).]).\N\N\NThe paper under review is another attempt at measuring the level op projectivity of a module, this time using proper classes of exact sequences. Let \(M\) be a unital \(R\)-module, \(R\) an associative ring with identity. The notation \(\pi^{-1}(M)\) is used for the class of all short exact sequences \(\mathbb{E}\) such that \(\mathrm{Hom}(M, \mathbb{E})\) is exact. The module \(M\) is called \(\pi\)-indigent if \(\pi^{-1}(M)\) consists of all splitting exact sequences. Any \(\pi\)-indigent module is shown to be both \(p\)-poor and \(p\)-indigent, but the converse is not true. The main theorem shows that, for a non-semisimple ring \(R\), every non-projective module is \(\pi\)-indigent if and only if every non-projective (finitely generated / finitely presented / cyclic / simple) module is \(\pi\)-indigent, if and only if \(R\) is an Artinian serial ring and \(J^{2}(R) = 0\) with a unique singular simple module (up to isomorphism). If \(R\) is not a right p.p.-ring, every non-projectve module is \(\pi\)-indigent if and only if every non-projective (right ideal / finitely generated right ideal / finitely presented right ideal / cyclic right ideal) of \(R\) is \(\pi\)-indigent. The final result proves that the existence of a non-projective indecomposable \(\pi\)-indigent \(R\)-module is equivalent to every non-projective (module with exchange property / (semi)simple module) being indigent.