C2 Property of Column Finite Matrix Rings

Abstract

A ring R is called a right C2 ring if any right ideal of R isomorphic to a direct summand of R R is also a direct summand. The ring R is called a right C3 ring if any sum of two independent summands of R R is also a direct summand. It is well known that a right C2 ring must be a right C3 ring but the converse assertion is not true. The ring R is called J -regular if R/J(R) is von Neumann regular, where J(R) is the Jacobson radical of R. Let ℕ be the set of natural numbers and let Λ be any infinite set. The following assertions are proved to be equivalent for a ring R: (1) ℂ $$ \mathbb{F}\mathbb{M} $$ ℕ R) is a right C2 ring; (2) ℂ $$ \mathbb{F}\mathbb{M} $$ Λ(R) is a right C2 ring; (3) ℂ $$ \mathbb{F}\mathbb{M} $$ ℕ(R) is a right C3 ring; (4) ℂ $$ \mathbb{F}\mathbb{M} $$ Λ(R) is a right C3 ring; (5) ℂ $$ \mathbb{F}\mathbb{M} $$ ℕ(R) is a J -regular ring and $$ \mathbb{M} $$ n (R) is a right C2 (or right C3) ring for all integers n ≥ 1.