Relating properties of a ring and its ring of row and column finite matrices (Q5952406)

The Mackey-Ornstein theorem, mentioned in \textit{I. Kaplansky}'s book ``Rings of operators'' (1968; Zbl 0174.18503), states that if \(R\) is a semisimple ring then the ring \(\text{RCFM}_\Gamma(R)\) of row and column finite matrices over \(R\) is a Baer ring for any infinite set \(\Gamma\). Here a ring with identity is a Baer ring if every left (equivalently every right) annihilator is generated by an idempotent. The authors prove that the converse is true, i.e., \(R\) must be semisimple if \(\text{RCFM}_\Gamma(R)\) is a Baer ring for some infinite set \(\Gamma\). The proof is long and the authors develop techniques to obtain more results for \(\text{RCFM}_\Gamma(R)\), where \(R\) is a perfect or semiprimary ring. Finally, they obtain results on annihilators in \(\text{RCFM}_\mathbb{N}(\mathbb{Z})\) to show that this ring is left and right coherent.