Relating properties of a ring with properties of matrix rings coming from Ornstein dual pairs (Q1398149)

The authors study the ring-theoretic classification of intermediate ring \({\mathcal E}_{\alpha\beta}(R)\) of infinite rings from Ornstein dual pairs in case \(R\) is semisimple Artinian, semiprimary or left or right perfect. They develop a technique of decomposing an infinite matrix as ``infinite sum of submatrices of less size''. Then, they show that \(R\) is a semisimple Artinian ring if and only if \({\mathcal E}_{\alpha\beta}(R)\) is a von Neumann regular ring. They then describe the lattice of finitely generated left ideals, and the lattice of two-sided ideals of \({\mathcal E}_{\alpha\beta}(R)\) by equivalence of idempotents. They also describe the Jacobson radical of \({\mathcal E}_{\alpha\beta}(R)\) for an arbitrary ring \(R\), and then they show among other results, that \(R\) is left perfect if and only if \({\mathcal E}_{\alpha\beta}(R)\) is a semiregular ring. The paper is divided into sections of Matrix rings over semisimple rings; Equivalence of idempotents; and Jacobson radical and semiregular matrix subrings.