C2 Property of Column Finite Matrix Rings - Chen, Jianlong; Shen, Liang (Ukrains’kyi Matematychnyi Zhurnal, 2014)
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.