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

Abstract

In this paper we study the ring-theoretic classification of intermediate rings, Eαβ(R) of infinite matrix rings, RCFMα(R)⊆Eαβ(R)⊆RFMα(R), from Ornstein's dual pairs (see [D. Ornstein, Dual vector spaces, Ann. Math. 69 (1959) 520–534; A. del Rio, J.J. Simón, Intermediate rings between matrix rings and Ornstein dual pairs, Arch. Math. (Basel) 75 (2000) 256–263]) in case R is semisimple artinian, semiprimary or left or right perfect. To do this, we develop a technique of decomposing an infinite matrix as “infinite sum of submatrices of less size.” Then, we show that R is a semisimple artinian ring if and only if Eαβ(R) is a von Neumann regular ring. We then describe the lattice of finitely generated left ideals, and the lattice of two-sided ideals of Eαβ(R) by equivalence of idempotents; that is, in classical terms. We also describe the Jacobson radical of Eαβ(R) for an arbitrary ring R, and then we show, among other results, that R is left perfect if and only if Eαβ(R) is a semiregular ring.