$C2$ 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 $R$ 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) $ℂFMFM_{ℕ} (R)$ is a right $C2$ ring;
(2) $ℂFMFM_{Λ}(R)$ is a right $C2$ ring;
(3) $ℂFMFM_{ℕ}(R)$ is a right $C3$ ring;
(4) $ℂFMFM_{Λ}(R)$ is a right $C3$ ring;
(5) $ℂFMFM_{ℕ}(R)$ is a $J$ -regular ring and $M_n(R)$ is a right $C2$ (or right $C3$) ring for all integers $n ≥ 1$.