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

✍ Scribed by Claudi Busqué; Juan Jacobo Simón

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 133 KB
• Volume: 264
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00133-9

✦ Synopsis

In this paper we study the ring-theoretic classification of intermediate rings, 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 E αβ (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 E αβ (R) by equivalence of idempotents; that is, in classical terms. We also describe the Jacobson radical of E αβ (R) for an arbitrary ring R, and then we show, among other results, that R is left perfect if and only if E αβ (R) is a semiregular ring.